Chapter 6 The Triangle and its Properties (Additional Questions)

(C)

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A triangle is a basic polygon in geometry. By definition, a triangle is a closed shape formed by three line segments connecting three non-collinear points.

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These three line segments are called the sides of the triangle, and the angles formed by the intersection of these sides at the vertices are called the angles of the triangle.

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Thus, a triangle has exactly three sides and three angles.

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Reviewing the options provided:

(A) 2 sides and 3 angles - Incorrect, a triangle has 3 sides.

(B) 3 sides and 2 angles - Incorrect, a triangle has 3 angles.

(C) 3 sides and 3 angles - Correct, matches the definition of a triangle.

(D) 4 sides and 4 angles - Incorrect, this describes a quadrilateral, not a triangle.

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Therefore, the correct answer is (C).

(C)

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Let's consider the definitions of the types of triangles given in the options based on their side lengths:

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(A) Scalene triangle: A triangle where all three sides have different lengths.

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(B) Isosceles triangle: A triangle where at least two sides have equal length. An equilateral triangle is a special case of an isosceles triangle where all three sides are equal.

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(C) Equilateral triangle: A triangle where all three sides have equal length. This is precisely the description given in the question.

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(D) Right-angled triangle: A triangle where one of the angles is a right angle ($90^\circ$). This classification is based on angles, not directly on side lengths, although the side lengths in a right-angled triangle are related by the Pythagorean theorem ($a^2 + b^2 = c^2$). A right-angled triangle can also be scalene or isosceles.

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Based on these definitions, a triangle with all three sides of equal length is called an equilateral triangle.

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Therefore, the correct option is (C).

(B)

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One of the fundamental properties of Euclidean geometry concerning triangles is the sum of their interior angles.

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For any triangle, regardless of its size or shape (e.g., equilateral, isosceles, scalene, right-angled, acute, obtuse), the sum of the measures of its three interior angles is always constant.

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This constant sum is $180^\circ$ (or $\pi$ radians).

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This property can be proven using principles of parallel lines and transversals.

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Let the three interior angles of a triangle be $\angle A$, $\angle B$, and $\angle C$. The property states:

$\angle A + \angle B + \angle C = 180^\circ$

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Looking at the given options:

(A) $90^\circ$ - Incorrect. This is the sum of angles in, for example, an isosceles right triangle (excluding the right angle) or one angle in a right triangle.

(B) $180^\circ$ - Correct. This is the established geometric property.

(C) $270^\circ$ - Incorrect. This value is larger than the sum of angles in a triangle.

(D) $360^\circ$ - Incorrect. This is the sum of interior angles in a quadrilateral (a four-sided polygon).

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Therefore, the sum of the interior angles of a triangle is always $180^\circ$. The correct option is (B).

(C)

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Let's consider a triangle ABC. If we extend one side, say BC, to a point D, the angle $\angle ACD$ is an exterior angle of the triangle at vertex C.

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The angle $\angle ACB$ is the adjacent interior angle to the exterior angle $\angle ACD$.

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The other two interior angles of the triangle, $\angle BAC$ (or $\angle A$) and $\angle ABC$ (or $\angle B$), are called the opposite interior angles with respect to the exterior angle $\angle ACD$.

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The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

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For triangle ABC with exterior angle $\angle ACD$ at vertex C:

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Let's examine the options based on this theorem:

(A) Adjacent interior angle and one opposite interior angle - Incorrect. It's the sum of the *two* opposite interior angles, not the adjacent one.

(B) Two adjacent interior angles - Incorrect. There is only one adjacent interior angle at a vertex; the other two are opposite.

(C) Two opposite interior angles - Correct. This matches the Exterior Angle Theorem.

(D) All three interior angles - Incorrect. The sum of all three interior angles is $180^\circ$, while an exterior angle is typically different from $180^\circ$ (unless the opposite interior angles are $0^\circ$, which is not possible for a triangle).

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Therefore, the exterior angle is equal to the sum of the two opposite interior angles. The correct option is (C).

(B)

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Triangles are classified based on their angles into three types:

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1. Acute-angled triangle: All three interior angles are less than $90^\circ$.

2. Right-angled triangle: Exactly one interior angle is equal to $90^\circ$.

3. Obtuse-angled triangle: Exactly one interior angle is greater than $90^\circ$.

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We are given a triangle with angles $50^\circ$, $60^\circ$, and $70^\circ$.

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Let's check each angle:

$50^\circ < 90^\circ$

$60^\circ < 90^\circ$

$70^\circ < 90^\circ$

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Since all three angles ($50^\circ$, $60^\circ$, and $70^\circ$) are less than $90^\circ$, the triangle is an acute-angled triangle.

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Let's also verify that the sum of the angles is $180^\circ$:

$50^\circ + 60^\circ + 70^\circ = 180^\circ$.

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Now let's consider the given options:

(A) Right-angled triangle - Incorrect, as no angle is $90^\circ$.

(B) Acute-angled triangle - Correct, as all angles are less than $90^\circ$.

(C) Obtuse-angled triangle - Incorrect, as no angle is greater than $90^\circ$.

(D) Isosceles triangle - Incorrect. An isosceles triangle has at least two equal angles. The given angles $50^\circ$, $60^\circ$, and $70^\circ$ are all different. Therefore, it is a scalene triangle based on side lengths (as angles opposite to equal sides are equal, and vice versa).

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The correct classification based on the angles is acute-angled.

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The final answer is (B).

(C)

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An isosceles triangle is defined as a triangle that has at least two sides of equal length.

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A fundamental property of isosceles triangles, known as the Isosceles Triangle Theorem, relates the lengths of the sides to the measures of the angles.

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The theorem states: If two sides of a triangle are equal, then the angles opposite those sides are also equal.

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Conversely, if two angles of a triangle are equal, then the sides opposite those angles are also equal.

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Let's denote a triangle as ABC. If side AB is equal to side AC (AB = AC), then the angle opposite side AB (which is $\angle C$) is equal to the angle opposite side AC (which is $\angle B$). So, $\angle B = \angle C$.

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Examining the options:

(A) Complementary: Their sum is $90^\circ$. This is not a general property for the base angles of an isosceles triangle.

(B) Supplementary: Their sum is $180^\circ$. This applies to angles on a straight line or consecutive interior angles between parallel lines, not the angles opposite equal sides in a triangle.

(C) Equal: They have the same measure. This is the correct property as stated by the Isosceles Triangle Theorem.

(D) $90^\circ$: Each angle is $90^\circ$. If two angles in a triangle were $90^\circ$, their sum would be $180^\circ$, leaving the third angle as $0^\circ$, which is impossible for a triangle.

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Therefore, in an isosceles triangle, if two sides are equal, the angles opposite to these sides are also equal.

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The correct option is (C).

(B)

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An equilateral triangle is defined as a triangle in which all three sides are of equal length.

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A property of triangles is that angles opposite to equal sides are equal. In an equilateral triangle, since all three sides are equal, the angles opposite to these sides must also be equal.

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Let the measure of each angle in the equilateral triangle be $x$. Since all three angles are equal, they are all $x$.

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The sum of the interior angles of any triangle is always $180^\circ$. Therefore, for an equilateral triangle:

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Simplifying the equation:

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Solving for $x$:

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Thus, each interior angle of an equilateral triangle measures $60^\circ$.

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Reviewing the options:

(A) $30^\circ$ - Incorrect.

(B) $60^\circ$ - Correct.

(C) $90^\circ$ - Incorrect. A triangle with a $90^\circ$ angle is a right-angled triangle.

(D) $120^\circ$ - Incorrect. This would make the sum of angles $> 180^\circ$.

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The correct measure for each angle in an equilateral triangle is $60^\circ$.

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The final answer is (B).

(A)

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For any set of three angles to form a valid triangle, their sum must be exactly $180^\circ$. This is the fundamental Angle Sum Property of a triangle.

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Let's find the sum of the given angles: $45^\circ$, $45^\circ$, and $90^\circ$.

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Since the sum of the given angles is $180^\circ$, a triangle with these angles can exist.

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Now, let's classify the triangle based on its angles:

• One angle is $90^\circ$. A triangle with one angle equal to $90^\circ$ is called a right-angled triangle.
• Two angles are equal ($45^\circ = 45^\circ$). In a triangle, if two angles are equal, the sides opposite to these angles are also equal. A triangle with at least two equal sides (and thus two equal angles) is called an isosceles triangle.

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Therefore, a triangle with angles $45^\circ$, $45^\circ$, and $90^\circ$ is both a right-angled triangle and an isosceles triangle. It is specifically a right-angled isosceles triangle.

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Reviewing the options:

(A) Yes, it is a right-angled isosceles triangle. - Correct, as determined by the sum of angles and angle properties.

(B) Yes, it is an acute-angled triangle. - Incorrect, as one angle is $90^\circ$ (not less than $90^\circ$).

(C) No, the sum of angles is not $180^\circ$. - Incorrect, as the sum is exactly $180^\circ$.

(D) No, it cannot be a triangle. - Incorrect, as the sum of angles is $180^\circ$, fulfilling the requirement for a triangle.

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The correct answer is that such a triangle can exist and it is a right-angled isosceles triangle.

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The final answer is (A).

(C)

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This question involves the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

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Let the lengths of the two given sides be $a = 5$ cm and $b = 7$ cm. Let the length of the third side be $c$. According to the Triangle Inequality Theorem, the following three conditions must be met for a triangle to exist:

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1. $a + b > c$

2. $a + c > b$

3. $b + c > a$

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Substituting the given values $a=5$ and $b=7$ into these inequalities:

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1. $5 + 7 > c \implies 12 > c$

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2. $5 + c > 7 \implies c > 7 - 5 \implies c > 2$

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3. $7 + c > 5 \implies c > 5 - 7 \implies c > -2$

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Since the length of a side must be a positive value, $c > -2$ is always true if $c > 0$. Combining the relevant inequalities ($12 > c$ and $c > 2$), we get the condition for the length of the third side:

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$2$ cm $ < c < 12$ cm

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Now, let's check the given options to see which value for $c$ falls within this range ($2 < c < 12$):

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(A) $c = 2$ cm: Is $2 < 2 < 12$? No, $2 \not< 2$. This violates the condition $c > 2$. (Specifically, $5+2 = 7$, which is not greater than the third side $7$).

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(B) $c = 12$ cm: Is $2 < 12 < 12$? No, $12 \not< 12$. This violates the condition $c < 12$. (Specifically, $5+7 = 12$, which is not greater than the third side $12$).

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(C) $c = 10$ cm: Is $2 < 10 < 12$? Yes, $2 < 10$ and $10 < 12$. This value satisfies both conditions.

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(D) $c = 13$ cm: Is $2 < 13 < 12$? No, $13 \not< 12$. This violates the condition $c < 12$. (Specifically, $5+7 = 12$, which is not greater than the third side $13$).

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Only the length 10 cm can be the length of the third side of the triangle, as it is the only value that satisfies the Triangle Inequality Theorem ($2 < 10 < 12$).

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The correct option is (C).

(C)

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A right-angled triangle is a triangle that contains one angle that measures exactly $90^\circ$. This $90^\circ$ angle is called the right angle.

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The sides of a right-angled triangle are given specific names:

• The two sides that form the right angle are called the legs. In some contexts, especially when dealing with trigonometry relative to one of the acute angles, these legs may be referred to as the base and the perpendicular (or opposite side and adjacent side).
• The side that is directly opposite the right angle is the longest side of the triangle. This side has a unique name: the hypotenuse.

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The question asks for the name of the side opposite to the right angle.

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Based on the definition, the side opposite the $90^\circ$ angle in a right-angled triangle is the hypotenuse.

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Let's look at the options:

(A) Base - Incorrect. The base is usually one of the legs.

(B) Perpendicular - Incorrect. The perpendicular is usually the other leg (relative to a chosen base and acute angle).

(C) Hypotenuse - Correct. This is the specific name for the side opposite the right angle.

(D) Altitude - Incorrect. An altitude is a line segment representing the height of the triangle from a vertex perpendicular to the opposite side (or its extension).

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Therefore, the side opposite to the right angle in a right-angled triangle is called the hypotenuse.

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The final answer is (C).

(A)

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The Pythagorean Theorem is a fundamental principle in geometry that applies specifically to right-angled triangles.

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In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs (often referred to as the base and perpendicular depending on orientation or the angle of reference).

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The Pythagorean Theorem states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the other two sides (the legs).

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Mathematically, if $c$ represents the length of the hypotenuse and $a$ and $b$ represent the lengths of the other two sides (legs), the theorem can be written as:

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In words, "the square of the hypotenuse is equal to the sum of the squares of the other two sides."

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Let's evaluate the options:

(A) Other two sides: This correctly refers to the two legs of the right-angled triangle, whose squares sum up to the square of the hypotenuse. The theorem states the sum of the squares of the *lengths* of these sides.

(B) Altitude and base: While the legs can sometimes be referred to as base and perpendicular (a form of altitude), the theorem applies specifically to the lengths of the sides forming the right angle, which are generally called the other two sides or legs. "Other two sides" is a more general and accurate term for the legs in the context of the theorem.

(C) Two smaller angles: The theorem relates the *lengths* of the sides, not the measures of the angles.

(D) All three sides: The theorem is a specific relationship between the square of one side (hypotenuse) and the sum of the squares of the other two sides, not a general relationship involving all three sides summed or squared together in a different manner.

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Therefore, the statement that the square of the hypotenuse is equal to the sum of the squares of the other two sides accurately describes the Pythagorean property.

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The correct option is (A).

(A)

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To determine if a triangle with given side lengths is a right-angled triangle, we use the Converse of the Pythagorean Theorem.

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The Converse of the Pythagorean Theorem states: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right-angled triangle, and the angle opposite the longest side is the right angle ($90^\circ$).

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The given side lengths are 3 cm, 4 cm, and 5 cm.

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Identify the longest side. The longest side is 5 cm. This side would be the potential hypotenuse if the triangle is right-angled.

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Let $a=3$, $b=4$, and $c=5$ (where $c$ is the longest side). We need to check if $a^2 + b^2 = c^2$.

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Calculate the sum of the squares of the two shorter sides:

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Calculate the square of the longest side:

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Comparing the results, we see that the sum of the squares of the two shorter sides ($25$) is equal to the square of the longest side ($25$).

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Since the condition of the Converse of the Pythagorean Theorem is satisfied, the triangle is indeed a right-angled triangle. The right angle is opposite the side of length 5 cm.

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Let's consider the options:

(A) Yes - Correct, as the sides satisfy the Pythagorean theorem.

(B) No - Incorrect.

(C) Cannot be determined - Incorrect, it can be determined using the Converse of the Pythagorean Theorem.

(D) Only if the angles are 90, 45, 45 - Incorrect. A triangle with angles 90, 45, 45 is a right-angled isosceles triangle. Its sides are in the ratio $x : x : x\sqrt{2}$. While it is a right-angled triangle, the condition for being right-angled based on side lengths 3, 4, 5 is checked specifically using the Pythagorean theorem, regardless of the specific angle values (other than the $90^\circ$). The angles of a 3-4-5 right triangle are approximately $36.87^\circ$, $53.13^\circ$, and $90^\circ$, not $45^\circ, 45^\circ, 90^\circ$.

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Therefore, a triangle with sides 3 cm, 4 cm, and 5 cm is a right-angled triangle.

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The final answer is (A).

(A)

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We can solve this problem using the property of the exterior angle of a triangle.

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The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

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In $\triangle\text{ABC}$, the exterior angle at vertex A is given as $120^\circ$. The interior angles opposite to vertex A are $\angle\text{B}$ and $\angle\text{C}$.

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According to the Exterior Angle Theorem:

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We are given that the exterior angle at A is $120^\circ$ and $\angle\text{B} = 50^\circ$. Substitute these values into the equation:

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Now, solve for $\angle\text{C}$:

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Alternatively, we can use the properties of linear pairs and the angle sum property of a triangle.

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Let the interior angle at vertex A be $\angle\text{BAC}$. The exterior angle at A and the interior angle $\angle\text{BAC}$ form a linear pair, so their sum is $180^\circ$.

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Now, apply the Angle Sum Property of a triangle, which states that the sum of the interior angles of a triangle is $180^\circ$.

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Substitute the values $\angle\text{BAC} = 60^\circ$ and $\angle\text{B} = 50^\circ$:

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Both methods yield the same result.

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Therefore, $\angle\text{C} = 70^\circ$.

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Comparing this result with the options:

(A) $70^\circ$ - Correct.

(B) $60^\circ$ - Incorrect.

(C) $50^\circ$ - Incorrect.

(D) $130^\circ$ - Incorrect.

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The correct option is (A).

(B)

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For any three angles to form a triangle, their sum must be exactly $180^\circ$. This is known as the Angle Sum Property of a triangle.

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Let's check the sum of the angles for each given option:

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(A) $50^\circ, 60^\circ, 80^\circ$

Since the sum ($190^\circ$) is not equal to $180^\circ$, these angles cannot form a triangle.

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(B) $70^\circ, 70^\circ, 50^\circ$

Since the sum ($180^\circ$) is equal to $180^\circ$, these angles can form a triangle. Specifically, it would be an isosceles triangle because two angles are equal.

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(C) $90^\circ, 40^\circ, 60^\circ$

Since the sum ($190^\circ$) is not equal to $180^\circ$, these angles cannot form a triangle.

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(D) $100^\circ, 40^\circ, 30^\circ$

Since the sum ($170^\circ$) is not equal to $180^\circ$, these angles cannot form a triangle.

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Only the set of angles in option (B) sums up to $180^\circ$.

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The correct option is (B).

(A)

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Let's match each type of triangle with its defining angle property:

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(i) Acute-angled triangle: This type of triangle has all three interior angles measuring less than $90^\circ$. This matches property (c).

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(ii) Obtuse-angled triangle: This type of triangle has exactly one interior angle measuring greater than $90^\circ$. This matches property (d).

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(iii) Right-angled triangle: This type of triangle has exactly one interior angle measuring equal to $90^\circ$. This matches property (a).

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(iv) Equilateral triangle: An equilateral triangle has all three sides equal. As a consequence, all three interior angles are also equal. Since the sum of the angles is $180^\circ$, each angle measures $180^\circ / 3 = 60^\circ$. Thus, all angles are $60^\circ$. This matches property (b).

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So the correct matching is:

(i) - (c)

(ii) - (d)

(iii) - (a)

(iv) - (b)

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Let's check the options provided against this matching:

(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b) - This matches our derived pairings.

(B) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b) - Incorrect (ii) is matched with (a) and (iii) with (d).

(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b) - Incorrect (i) is matched with (d) and (ii) with (c).

(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a) - Incorrect (iii) is matched with (b) and (iv) with (a).

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The correct option is (A).

(D)

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Let's analyze the Assertion (A): "A triangle can have two obtuse angles."

An obtuse angle is an angle whose measure is greater than $90^\circ$. If a triangle were to have two obtuse angles, say $\angle_1$ and $\angle_2$, then $\angle_1 > 90^\circ$ and $\angle_2 > 90^\circ$.

The sum of these two angles would be $\angle_1 + \angle_2 > 90^\circ + 90^\circ = 180^\circ$.

However, the sum of all three interior angles of a triangle must be exactly $180^\circ$. If two angles already sum to more than $180^\circ$, the third angle would have to be negative ($180^\circ - (\angle_1 + \angle_2) < 180^\circ - 180^\circ = 0^\circ$), which is not possible for a triangle.

Therefore, a triangle cannot have two obtuse angles. The Assertion (A) is false.

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Now let's analyze the Reason (R): "The sum of angles in a triangle is $180^\circ$, and two obtuse angles would sum to more than $180^\circ$."

The first part of the reason states: "The sum of angles in a triangle is $180^\circ$." This is a fundamental and correct property of Euclidean geometry (the Angle Sum Property of a triangle).

The second part states: "two obtuse angles would sum to more than $180^\circ$." As explained above, if two angles are both greater than $90^\circ$, their sum will indeed be greater than $180^\circ$. This statement is also correct.

Since both parts of the reason are true, the Reason (R) statement as a whole is true.

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We have determined that Assertion (A) is false and Reason (R) is true.

Let's check the options based on this finding:

(A) Both A and R are true... - Incorrect, as A is false.

(B) Both A and R are true... - Incorrect, as A is false.

(C) A is true, but R is false. - Incorrect, as A is false and R is true.

(D) A is false, but R is true. - Correct, this matches our findings.

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Furthermore, the Reason (R) correctly explains *why* the Assertion (A) is false. The fact that two obtuse angles sum to more than $180^\circ$ directly contradicts the Angle Sum Property of a triangle ($180^\circ$), thus making it impossible for a triangle to have two obtuse angles.

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The final answer is (D).

(A)

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Let's analyze the Assertion (A): "A triangle with side lengths 6 cm, 8 cm, and 10 cm is a right-angled triangle."

To check if a triangle is right-angled given its side lengths, we use the Converse of the Pythagorean Theorem. This theorem states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.

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The given side lengths are 6 cm, 8 cm, and 10 cm. The longest side is 10 cm.

Let's calculate the square of the longest side and the sum of the squares of the other two sides:

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Since the square of the longest side ($100$) is equal to the sum of the squares of the other two sides ($100$), the triangle satisfies the Converse of the Pythagorean Theorem. Therefore, the triangle with side lengths 6 cm, 8 cm, and 10 cm is indeed a right-angled triangle.

Assertion (A) is true.

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Now let's analyze the Reason (R): "In a right-angled triangle, the square of the largest side is equal to the sum of the squares of the other two sides ($6^2 + 8^2 = 36 + 64 = 100 = 10^2$)."

The first part of the reason states the Pythagorean Theorem, which is a true statement about right-angled triangles.

The second part performs the calculation for the given side lengths ($6^2 + 8^2 = 36 + 64 = 100 = 10^2$) and shows that they satisfy the Pythagorean relationship. This calculation is correct.

Reason (R) is true.

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Finally, let's check if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) claims the triangle is right-angled. Reason (R) provides the Pythagorean Theorem and demonstrates that the given sides satisfy this theorem. The Converse of the Pythagorean Theorem (which is essentially what R describes and shows) is the exact reason why a triangle with sides 6, 8, and 10 is a right-angled triangle.

Therefore, Reason (R) is the correct explanation of Assertion (A).

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Comparing our findings with the options:

(A) Both A and R are true, and R is the correct explanation of A. - This matches our conclusion.

(B) Both A and R are true, but R is not the correct explanation of A. - Incorrect.

(C) A is true, but R is false. - Incorrect.

(D) A is false, but R is true. - Incorrect.

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The final answer is (A).

(D)

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Context:

The situation described involves a ladder leaning against a wall, with the ground below. Assuming the wall is perpendicular to the ground, this setup forms a right-angled triangle. The components of this triangle are:

• The ground from the base of the wall to the foot of the ladder (length = 3 meters) - This is one leg of the right-angled triangle.
• The wall from the ground up to where the ladder touches (height = 4 meters) - This is the other leg of the right-angled triangle.
• The ladder itself - This is the side opposite the right angle, which is the hypotenuse.

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Problem: We need to find the length of the ladder (the hypotenuse).

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Property Used:

The relationship between the lengths of the sides in a right-angled triangle is defined by the Pythagorean Theorem (also known as Pythagoras property). The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

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If we let the lengths of the legs be $a$ and $b$, and the length of the hypotenuse be $c$, the theorem is expressed as:

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In this specific case, the lengths of the legs are given as 3 meters and 4 meters. We can use the Pythagoras property to find the length of the hypotenuse ($c$), which represents the length of the ladder:

Thus, the length of the ladder is 5 meters, calculated using the Pythagoras property.

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Let's consider why the other options are not the primary property used to find the length in this specific scenario:

• (A) Angle sum property of a triangle: While true ($180^\circ$), this property relates angles and does not directly provide a way to calculate side lengths given other side lengths.
• (B) Exterior angle property: This relates an exterior angle to the sum of interior angles. It does not directly relate the lengths of the sides in this manner.
• (C) Sum of lengths of two sides property (Triangle Inequality): This property ($a+b > c$) determines if a triangle can exist with given side lengths. It gives a range for the possible length of the third side but does not provide a formula to calculate its exact value given the other two sides in a right triangle.

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Therefore, the property used to find the length of the ladder in this right-angled triangle scenario is the Pythagoras property.

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The correct option is (D).

(A)

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As established in the analysis of Question 18's case study, the situation involving the ladder, the wall, and the ground forms a right-angled triangle.

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The given information provides the lengths of the two legs of this right-angled triangle:

• Distance from the wall to the foot of the ladder (one leg) = 3 meters.
• Height the ladder reaches on the wall (the other leg) = 4 meters.

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The length of the ladder is the length of the hypotenuse of this right-angled triangle (the side opposite the $90^\circ$ angle between the wall and the ground).

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According to the Pythagoras property (Pythagorean Theorem), in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides (legs, $a$ and $b$).

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Let $a = 3$ meters and $b = 4$ meters. We need to find $c$, the length of the ladder.

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Substitute the values into the formula:

---

To find the length $c$, we take the square root of both sides:

Since length must be a positive value, we take the positive square root.

---

The length of the ladder is 5 meters.

---

Let's compare this result with the given options:

(A) 5 meters - Correct.

(B) 7 meters - Incorrect ($3+4=7$, but the theorem is about squares).

(C) $\sqrt{7}$ meters - Incorrect ($3+4=7$, taking square root of sum of sides, not sum of squares).

(D) $\sqrt{12}$ meters - Incorrect ($3 \times 4=12$, taking square root of product, not sum of squares).

---

The length of the ladder is 5 meters.

---

The final answer is (A).

(B)

---

We are given a triangle $\triangle\text{PQR}$ where $\angle\text{P} = \angle\text{Q}$.

---

A fundamental property of triangles relates the angles to the lengths of the sides opposite them. The Converse of the Isosceles Triangle Theorem states that if two angles of a triangle are equal, then the sides opposite those angles are also equal in length.

---

In $\triangle\text{PQR}$:

• The side opposite angle $\angle\text{P}$ is side QR.
• The side opposite angle $\angle\text{Q}$ is side PR.

---

Since $\angle\text{P} = \angle\text{Q}$, according to the Converse of the Isosceles Triangle Theorem, the sides opposite these angles must be equal.

---

A triangle with at least two sides of equal length is defined as an isosceles triangle.

---

Let's examine the options based on this classification:

(A) Scalene: A scalene triangle has all three sides of different lengths. This contradicts our finding that PR = QR.

(B) Isosceles: An isosceles triangle has at least two sides of equal length. Since we found PR = QR, the triangle fits this definition.

(C) Equilateral: An equilateral triangle has all three sides of equal length (PQ = QR = RP). This would imply all three angles are equal ($\angle\text{P} = \angle\text{Q} = \angle\text{R} = 60^\circ$). While an equilateral triangle is a special case of an isosceles triangle, the condition $\angle\text{P} = \angle\text{Q}$ does not guarantee that $\angle\text{R}$ is also equal to $\angle\text{P}$ and $\angle\text{Q}$. For example, a triangle with angles $70^\circ, 70^\circ, 40^\circ$ has two equal angles ($\angle\text{P}=\angle\text{Q}=70^\circ$), making it isosceles, but it is not equilateral.

(D) Right-angled: A right-angled triangle has one angle equal to $90^\circ$. The condition $\angle\text{P} = \angle\text{Q}$ does not provide any information about whether any angle is $90^\circ$. For instance, an isosceles triangle could have angles $50^\circ, 50^\circ, 80^\circ$ (acute-angled) or $20^\circ, 20^\circ, 140^\circ$ (obtuse-angled), neither of which are right-angled.

---

Based on the property that sides opposite equal angles are equal, if $\angle\text{P} = \angle\text{Q}$, then the sides PR and QR are equal, making $\triangle\text{PQR}$ an isosceles triangle.

---

The correct option is (B).

(D)

---

Triangles are typically classified in two main ways: by their side lengths and by their angle measures.

---

Classification of Triangles by Sides:

• Scalene triangle: A triangle in which all three sides have different lengths.
• Isosceles triangle: A triangle in which at least two sides have equal lengths.
• Equilateral triangle: A triangle in which all three sides have equal lengths. (An equilateral triangle is a special type of isosceles triangle).

---

Classification of Triangles by Angles:

• Acute-angled triangle: A triangle in which all three angles are less than $90^\circ$.
• Right-angled triangle: A triangle in which one angle is exactly $90^\circ$.
• Obtuse-angled triangle: A triangle in which one angle is greater than $90^\circ$.

---

Now let's look at the given options:

• (A) Scalene: This is a classification based on side lengths.
• (B) Isosceles: This is a classification based on side lengths.
• (C) Equilateral: This is a classification based on side lengths.
• (D) Obtuse: This term refers to an angle greater than $90^\circ$. When used to describe a triangle, it implies an obtuse-angled triangle, which is a classification based on its angle measures, not its side lengths.

---

Therefore, "Obtuse" is the option that does NOT represent a classification of a triangle based on its sides.

---

The correct option is (D).

(C)

---

Given:

An isosceles triangle.

Vertex angle = $80^\circ$.

---

To Find:

Measure of each base angle.

---

Solution:

In an isosceles triangle, the two base angles (the angles opposite the equal sides) are equal.

---

Let the measure of each base angle be $x^\circ$.

The vertex angle is given as $80^\circ$.

---

The sum of the interior angles of any triangle is always $180^\circ$.

---

Using the Angle Sum Property for the given isosceles triangle:

---

Substitute the given and assumed values:

---

Combine like terms:

---

Subtract $80^\circ$ from both sides of the equation:

---

Divide by 2 to solve for $x$:

---

So, the measure of each base angle is $50^\circ$.

---

Let's verify this: $80^\circ + 50^\circ + 50^\circ = 80^\circ + 100^\circ = 180^\circ$, which is correct for a triangle.

---

Comparing with the given options:

(A) $80^\circ$ - Incorrect.

(B) $100^\circ$ - Incorrect.

(C) $50^\circ$ - Correct.

(D) $20^\circ$ - Incorrect.

---

The correct option is (C).

(C)

---

Given:

The angles of a triangle are in the ratio $1:2:3$.

---

To Find:

The type of triangle.

---

Solution:

Let the measures of the three angles of the triangle be $x^\circ$, $2x^\circ$, and $3x^\circ$, based on the given ratio $1:2:3$.

---

The sum of the interior angles of any triangle is always $180^\circ$.

---

Using the Angle Sum Property of a triangle:

---

Combine the terms on the left side:

---

Solve for $x$ by dividing both sides by 6:

---

Now, find the measure of each angle by substituting the value of $x$:

First angle $= x^\circ = 30^\circ$

Second angle $= 2x^\circ = 2 \times 30^\circ = 60^\circ$

Third angle $= 3x^\circ = 3 \times 30^\circ = 90^\circ$

---

The angles of the triangle are $30^\circ$, $60^\circ$, and $90^\circ$.

---

Now, classify the triangle based on its angles:

• Since one of the angles is exactly $90^\circ$, the triangle is a right-angled triangle.
• It is not acute-angled because one angle is $90^\circ$ (not less than $90^\circ$).
• It is not obtuse-angled because no angle is greater than $90^\circ$.
• It is not equilateral because the angles are not all equal ($60^\circ$).

---

Thus, a triangle with angles in the ratio $1:2:3$ is a right-angled triangle.

---

Comparing this with the options:

(A) Acute-angled - Incorrect.

(B) Obtuse-angled - Incorrect.

(C) Right-angled - Correct.

(D) Equilateral - Incorrect.

---

The correct option is (C).

(D)

---

The Triangle Inequality Theorem is a fundamental theorem in geometry that describes a necessary condition for three line segments to form a triangle.

---

The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

---

Given the lengths of the three sides of a triangle are $a$, $b$, and $c$, the Triangle Inequality Theorem requires that the following three inequalities must simultaneously hold true:

---

1. The sum of side lengths $a$ and $b$ must be greater than the length of side $c$:

$$a + b > c$$

---

2. The sum of side lengths $a$ and $c$ must be greater than the length of side $b$:

$$a + c > b$$

---

3. The sum of side lengths $b$ and $c$ must be greater than the length of side $a$:

$$b + c > a$$

---

For a set of three lengths to form a valid triangle, all three of these conditions must be satisfied.

---

Looking at the options provided:

(A) $a+b > c$ - This is one of the required conditions.

(B) $a+c > b$ - This is another one of the required conditions.

(C) $b+c > a$ - This is the third required condition.

(D) All of the above - This option correctly indicates that all three inequalities listed in (A), (B), and (C) must be true according to the Triangle Inequality Theorem.

---

Therefore, if the lengths of the sides of a triangle are $a$, $b$, and $c$, all three inequalities $a+b > c$, $a+c > b$, and $b+c > a$ must hold true. The correct option is (D).

(C)

---

The statement provided is the definition of the Triangle Inequality Theorem.

---

This theorem is a fundamental property in geometry that describes the condition necessary for three given lengths to form a valid triangle.

---

The theorem explicitly states:

"The sum of the lengths of any two sides of a triangle is always greater than the length of the third side."

---

If the sum of the lengths of two sides were equal to the length of the third side, the three points would be collinear (lie on a straight line), forming a degenerate triangle, not a true triangle with interior angles.

If the sum of the lengths of two sides were less than the length of the third side, the two shorter sides would not be long enough to meet, and thus would not form a closed shape (a triangle).

---

Let's consider the options to complete the statement:

(A) Equal to - Incorrect. This would mean the three points are collinear.

(B) Less than - Incorrect. The two sides would not meet to form a triangle.

(C) Greater than - Correct. This is the precise statement of the theorem.

(D) Less than or equal to - Incorrect. It must be strictly greater than.

---

Therefore, the correct word to complete the statement is "greater".

---

The completed statement is: "The sum of the lengths of any two sides of a triangle is always greater than the length of the third side."

---

The correct option is (C).

(A)

---

The Pythagoras property, or the Pythagorean Theorem, is a fundamental principle that applies to right-angled triangles.

---

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The other two sides, which form the right angle, are called the legs.

---

The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

---

Given that the legs have lengths $a$ and $b$, and the hypotenuse has length $c$, the mathematical statement of the Pythagoras property is:

---

Let's look at the options:

• (A) $a^2 + b^2 = c^2$: This correctly represents the sum of the squares of the legs equaling the square of the hypotenuse.
• (B) $a^2 + c^2 = b^2$: This implies the hypotenuse is $b$ and the legs are $a$ and $c$, which contradicts the given information that $c$ is the hypotenuse.
• (C) $b^2 + c^2 = a^2$: This implies the hypotenuse is $a$ and the legs are $b$ and $c$, also contradicting the given information.
• (D) $a+b=c$: This is related to the Triangle Inequality Theorem, specifically representing a degenerate triangle (where the three points are collinear). It is not the Pythagoras property, which deals with the relationship between the squares of the sides in a right-angled triangle.

---

Therefore, the correct statement according to Pythagoras property, for a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$, is $a^2 + b^2 = c^2$.

---

The correct option is (A).

(B)

---

To determine if a triangle can be formed with given side lengths, we use the Triangle Inequality Theorem.

---

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

---

Let the given side lengths be $a = 2$ cm, $b = 3$ cm, and $c = 6$ cm.

---

We need to check if the following three inequalities hold true:

---

1. $a + b > c$

This inequality is false.

---

Since the first condition of the Triangle Inequality Theorem is not met, we can immediately conclude that a triangle cannot be formed with these side lengths. We don't need to check the other two inequalities, but let's do so for completeness:

---

2. $a + c > b$

This inequality is true.

---

3. $b + c > a$

This inequality is true.

---

Although two of the conditions are met, the Triangle Inequality Theorem requires that all three conditions must be true. Since $2 + 3$ is not greater than 6, the sides of lengths 2 cm, 3 cm, and 6 cm cannot form a triangle.

---

Comparing with the options:

(A) Yes - Incorrect.

(B) No - Correct, as the Triangle Inequality Theorem is violated.

(C) Cannot be determined - Incorrect, it can be determined using the Triangle Inequality Theorem.

(D) Only if it's a right triangle - Incorrect. The ability to form a triangle is checked by the Triangle Inequality Theorem, regardless of whether it would be a right triangle (which it isn't, as $2^2 + 3^2 = 4 + 9 = 13 \neq 6^2 = 36$).

---

Therefore, a triangle cannot have sides of lengths 2 cm, 3 cm, and 6 cm.

---

The final answer is (B).

(A)

---

Given:

The angles of a triangle are $x^\circ$, $(x+10)^\circ$, and $(x+20)^\circ$.

---

To Find:

The value of $x$.

---

Solution:

The sum of the interior angles of any triangle is always $180^\circ$. This is known as the Angle Sum Property of a triangle.

---

Using the Angle Sum Property, we can write an equation:

---

Now, let's solve this linear equation for $x$. First, remove the parentheses and combine like terms on the left side:

---

Subtract 30 from both sides of the equation:

---

Divide both sides by 3 to find the value of $x$:

---

So, the value of $x$ is 50.

---

The angles of the triangle are therefore:

• $x = 50^\circ$
• $x+10^\circ = 50^\circ + 10^\circ = 60^\circ$
• $x+20^\circ = 50^\circ + 20^\circ = 70^\circ$

Checking the sum: $50^\circ + 60^\circ + 70^\circ = 110^\circ + 70^\circ = 180^\circ$. The sum is correct.

---

Comparing our result with the given options:

(A) $50^\circ$ - Correct.

(B) $60^\circ$ - Incorrect.

(C) $70^\circ$ - Incorrect.

(D) $80^\circ$ - Incorrect.

---

The correct option is (A).

(A)

---

Given:

In $\triangle\text{XYZ}$, the exterior angle at vertex Y is $150^\circ$.

One interior angle, $\angle\text{X} = 70^\circ$.

---

To Find:

The measure of angle $\angle\text{Z}$.

---

Solution:

We can use the Exterior Angle Theorem. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

---

In $\triangle\text{XYZ}$, the exterior angle at Y is $150^\circ$. The two interior angles opposite to vertex Y are $\angle\text{X}$ and $\angle\text{Z}$.

---

According to the Exterior Angle Theorem:

---

Substitute the given values into the equation:

---

To find $\angle\text{Z}$, subtract $70^\circ$ from both sides of the equation:

---

Alternate Method:

First, find the interior angle at vertex Y, denoted as $\angle\text{XYZ}$. The interior angle and the exterior angle at the same vertex form a linear pair, so their sum is $180^\circ$.

---

Now, use the Angle Sum Property of a triangle, which states that the sum of the interior angles of $\triangle\text{XYZ}$ is $180^\circ$.

---

Substitute the known values $\angle\text{X} = 70^\circ$ and $\angle\text{XYZ} = 30^\circ$:

---

Solve for $\angle\text{Z}$:

---

Both methods confirm that the measure of angle $\angle\text{Z}$ is $80^\circ$.

---

Comparing with the given options:

(A) $80^\circ$ - Correct.

(B) $150^\circ$ - Incorrect.

(C) $70^\circ$ - Incorrect.

(D) $100^\circ$ - Incorrect.

---

The final answer is (A).

(C)

---

Triangles are classified based on the lengths of their sides as follows:

---

• Scalene triangle: All three sides have different lengths.
• Isosceles triangle: At least two sides have equal lengths.
• Equilateral triangle: All three sides have equal lengths. (Note that an equilateral triangle is a special case of an isosceles triangle because it also has at least two equal sides).

---

Let's examine the given options in relation to the definition of an isosceles triangle:

• (A) All sides are equal: This describes an equilateral triangle, not an isosceles triangle in general (though it's a specific type).
• (B) All angles are $60^\circ$: This is a property of an equilateral triangle (since all sides are equal, all angles are equal, and the sum must be $180^\circ$). It's not true for all isosceles triangles (e.g., a triangle with angles $50^\circ, 50^\circ, 80^\circ$ is isosceles but not equilateral).
• (C) At least two sides are equal: This is the definition of an isosceles triangle.
• (D) One angle is $90^\circ$: This describes a right-angled triangle. An isosceles triangle can be right-angled (e.g., angles $45^\circ, 45^\circ, 90^\circ$), but not all isosceles triangles are right-angled (e.g., a triangle with angles $50^\circ, 50^\circ, 80^\circ$).

---

Therefore, the statement that is true for an isosceles triangle by definition is that at least two sides are equal.

---

The correct option is (C).

(C)

---

Given:

A square field.

Length of the diagonal = 13 meters.

---

To Find:

The length of the side of the square.

---

Solution:

A square has four equal sides and four right angles ($90^\circ$). When a diagonal is drawn in a square, it divides the square into two congruent right-angled triangles.

---

Let the side length of the square be $s$ meters.

In one of the right-angled triangles formed by the diagonal, the two legs are the sides of the square, each of length $s$. The hypotenuse is the diagonal of the square, which is given as 13 meters.

---

According to the Pythagoras property (Pythagorean Theorem), in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs).

---

Applying the Pythagoras property to the right-angled triangle formed by the diagonal:

---

Combine the terms on the left side:

---

Divide both sides by 2:

---

Take the square root of both sides to find $s$. Since length must be positive, we take the positive square root:

---

The length of the side of the square is $\frac{13}{\sqrt{2}}$ meters.

---

Comparing this result with the given options:

(A) $\sqrt{13}$ meters - Incorrect.

(B) 13 meters - Incorrect. This is the diagonal length.

(C) $\frac{13}{\sqrt{2}}$ meters - Correct.

(D) $13\sqrt{2}$ meters - Incorrect. This would be the diagonal length if the side was 13 meters.

---

The final answer is (C).

(B)

---

Given:

In $\triangle\text{ABC}$, $\angle\text{A} = 90^\circ$.

Length of side AB = 5 cm.

Length of side AC = 12 cm.

---

To Find:

The length of side BC.

---

Solution:

Since $\angle\text{A} = 90^\circ$, $\triangle\text{ABC}$ is a right-angled triangle with the right angle at vertex A.

---

In a right-angled triangle, the side opposite the right angle is the hypotenuse. The angle at A is $90^\circ$, so the side opposite to angle A is BC. Therefore, BC is the hypotenuse.

---

The other two sides, AB and AC, are the legs of the right-angled triangle.

---

We can use the Pythagoras property (Pythagorean Theorem) to find the length of the hypotenuse. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

---

Let AB $= a = 5$ cm, AC $= b = 12$ cm, and BC $= c$.

Applying the Pythagoras property:

---

Substitute the given values for AB and AC:

---

Calculate the squares and sum them:

---

To find the length of BC, take the square root of both sides. Since length is positive, we take the positive square root:

The length of the hypotenuse BC is 13 cm.

---

Comparing this result with the given options:

(A) 7 cm - Incorrect ($12-5=7$, but this is not the correct calculation).

(B) 13 cm - Correct.

(C) 17 cm - Incorrect ($5+12=17$, but the theorem is about squares).

(D) $\sqrt{119}$ cm - Incorrect ($12^2 - 5^2 = 144 - 25 = 119$, $\sqrt{119}$ would be the length of a leg if 12 was the hypotenuse and 5 was the other leg, but here 12 and 5 are legs).

---

The length of BC is 13 cm.

---

The final answer is (B).

(C)

---

Triangles are classified based on the measures of their interior angles. The common classifications are:

---

• Acute-angled triangle: A triangle in which all three interior angles are less than $90^\circ$ (i.e., all angles are acute).
• Right-angled triangle: A triangle in which exactly one interior angle is equal to $90^\circ$ (a right angle). The other two angles must be acute.
• Obtuse-angled triangle: A triangle in which exactly one interior angle is greater than $90^\circ$ (an obtuse angle). The other two angles must be acute.

---

The question asks which type of triangle *must* have all three angles acute.

---

Looking at the definitions:

• An obtuse-angled triangle has one obtuse angle and two acute angles. Not all three are acute.
• A right-angled triangle has one right angle and two acute angles. Not all three are acute.
• An acute-angled triangle has all three angles less than $90^\circ$, meaning all three angles are acute. This matches the requirement.

---

Therefore, the type of triangle that must have all three angles acute is an acute-angled triangle.

---

Comparing with the given options:

(A) Obtuse-angled triangle - Incorrect.

(B) Right-angled triangle - Incorrect.

(C) Acute-angled triangle - Correct.

(D) Cannot be determined - Incorrect.

---

The correct option is (C).

(C)

---

To determine if a triangle can be formed with three given side lengths, we use the Triangle Inequality Theorem.

---

The theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

---

For three lengths $a, b, c$ to form a triangle, the following three conditions must all be true:

$a + b > c$

$a + c > b$

$b + c > a$

---

Let's check each option:

---

(A) Side lengths: 4, 5, 6

Check the sum of the two smaller sides against the largest side: $4 + 5 = 9$.

Is $9 > 6$? Yes, $9 > 6$ is true.

(Also checking other pairs: $4+6 = 10 > 5$ True, $5+6 = 11 > 4$ True. All conditions hold.)

A triangle can be formed with these side lengths.

---

(B) Side lengths: 3, 3, 3

Check the sum of any two sides against the third side: $3 + 3 = 6$.

Is $6 > 3$? Yes, $6 > 3$ is true.

(This is an equilateral triangle, where all sums will be greater than the third side.)

A triangle can be formed with these side lengths.

---

(C) Side lengths: 1, 2, 3

Check the sum of the two smaller sides against the largest side: $1 + 2 = 3$.

Is $3 > 3$? No, $3 > 3$ is false. $3 = 3$.

Since the sum of two sides is equal to the third side, these lengths would form a degenerate triangle (the three points would lie on a straight line).

A proper triangle cannot be formed with these side lengths.

---

(D) Side lengths: 7, 8, 9

Check the sum of the two smaller sides against the largest side: $7 + 8 = 15$.

Is $15 > 9$? Yes, $15 > 9$ is true.

(Also checking other pairs: $7+9 = 16 > 8$ True, $8+9 = 17 > 7$ True. All conditions hold.)

A triangle can be formed with these side lengths.

---

The set of side lengths that violates the Triangle Inequality Theorem is 1, 2, and 3 because the sum of the two shorter sides (1+2=3) is not greater than the longest side (3).

---

Therefore, a triangle cannot be formed with side lengths 1, 2, and 3.

---

The correct option is (C).

(B)

---

Given:

Two angles of a triangle are $40^\circ$ and $50^\circ$.

---

To Find:

The measure of the third angle and the type of triangle based on angles.

---

Solution:

Let the three angles of the triangle be $\angle_1$, $\angle_2$, and $\angle_3$. We are given $\angle_1 = 40^\circ$ and $\angle_2 = 50^\circ$.

---

The sum of the interior angles of any triangle is always $180^\circ$. This is the Angle Sum Property of a triangle.

---

Substitute the given angle measures into the equation:

---

Combine the known angles:

---

Subtract $90^\circ$ from both sides to find the measure of the third angle:

---

So, the measure of the third angle is $90^\circ$. The three angles of the triangle are $40^\circ$, $50^\circ$, and $90^\circ$.

---

Now, we classify the triangle based on its angles:

• An acute-angled triangle has all three angles less than $90^\circ$. This triangle has a $90^\circ$ angle, so it is not acute-angled.
• A right-angled triangle has exactly one angle equal to $90^\circ$. This triangle has a $90^\circ$ angle.
• An obtuse-angled triangle has exactly one angle greater than $90^\circ$. This triangle has no angle greater than $90^\circ$.

---

Since the triangle has one angle measuring $90^\circ$, it is a right-angled triangle.

---

The third angle is $90^\circ$, and the triangle is right-angled.

---

Comparing this result with the given options:

(A) $90^\circ$, Acute-angled - Incorrect (the type is wrong).

(B) $90^\circ$, Right-angled - Correct.

(C) $100^\circ$, Obtuse-angled - Incorrect (the third angle is wrong, and hence the type based on that angle would be wrong).

(D) $90^\circ$, Equilateral - Incorrect (the type is wrong; an equilateral triangle has three $60^\circ$ angles).

---

The correct option is (B).

(B)

---

To answer this question, we need to recall the property relating the exterior angle of a triangle to its interior opposite angles. This property is known as the Exterior Angle Theorem.

---

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

---

Let's consider a triangle with interior angles $\angle A$, $\angle B$, and $\angle C$. If we extend one side, say AC to a point D, then the exterior angle at C is $\angle BCD$. The interior opposite angles to $\angle BCD$ are $\angle A$ and $\angle B$.

---

According to the Exterior Angle Theorem:

---

In any triangle, the measures of the interior angles must be positive values (greater than $0^\circ$). So, $\angle A > 0^\circ$ and $\angle B > 0^\circ$.

---

Since $\angle BCD$ is the sum of $\angle A$ and $\angle B$, and both $\angle A$ and $\angle B$ are positive, it follows that $\angle BCD$ must be greater than $\angle A$ and also greater than $\angle B$.

---

This shows that the exterior angle is always greater than *either* of the interior opposite angles.

---

Therefore, an exterior angle of a triangle cannot be less than either of the interior opposite angles.

---

Let's evaluate the options:

(A) Yes - Incorrect, as shown by the Exterior Angle Theorem.

(B) No - Correct, the exterior angle is always greater than each opposite interior angle.

(C) Only if the triangle is obtuse-angled - Incorrect. The Exterior Angle Theorem holds for all types of triangles (acute, right, or obtuse).

(D) Only if the triangle is acute-angled - Incorrect. The Exterior Angle Theorem holds for all types of triangles.

---

The correct option is (B).

(D)

---

The question asks for the property used to determine if three given line segments can form a triangle.

---

Let's review the properties mentioned in the options:

---

(A) Angle sum property: This property states that the sum of the interior angles of a triangle is $180^\circ$. This property is used to find an unknown angle in an existing triangle or to determine if a set of angles can form a triangle, but not directly to check if three given *line segments* can form a triangle.

---

(B) Exterior angle property: This property relates an exterior angle to the sum of the two opposite interior angles in an existing triangle. It is not used to check if three side lengths can form a triangle.

---

(C) Pythagoras property: This property relates the side lengths in a *right-angled* triangle ($a^2 + b^2 = c^2$). While it deals with side lengths, it is specific to right triangles and doesn't provide the general condition for *any* triangle to be formed from given side lengths.

---

(D) Sum of the lengths of two sides property: This is known as the Triangle Inequality Theorem. It states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. This theorem provides the necessary and sufficient condition for three given line segments to form a triangle.

---

If the lengths of the three line segments are $a$, $b$, and $c$, they can form a triangle if and only if:

$$a + b > c$$

$$a + c > b$$

$$b + c > a$$

---

Since this property directly allows us to determine if a triangle can be formed from given side lengths, it is the correct answer.

---

The correct option is (D).

A triangle can be defined based on its side lengths by comparing the lengths of its three sides. This comparison allows us to categorize triangles into specific types.

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Based on the lengths of their sides, triangles are classified into three main types:

1. Equilateral Triangle:

An equilateral triangle is a triangle in which all three sides are equal in length.

If the side lengths are $a$, $b$, and $c$, then for an equilateral triangle, $a = b = c$.

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2. Isosceles Triangle:

An isosceles triangle is a triangle in which at least two sides are equal in length.

If the side lengths are $a$, $b$, and $c$, then for an isosceles triangle, at least two of the sides must be equal (e.g., $a = b$ or $b = c$ or $a = c$). An equilateral triangle is a special case of an isosceles triangle.

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3. Scalene Triangle:

A scalene triangle is a triangle in which all three sides have different lengths.

If the side lengths are $a$, $b$, and $c$, then for a scalene triangle, $a \neq b$, $b \neq c$, and $a \neq c$.

A triangle can be defined based on its angle measures by examining the size of its three interior angles. The sum of the interior angles in any triangle is always $180^\circ$. Comparing each angle to $90^\circ$ allows us to categorize triangles into specific types.

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Based on the measures of their interior angles, triangles are classified into three main types:

1. Acute-angled Triangle (or Acute Triangle):

An acute-angled triangle is a triangle in which all three interior angles are acute angles. An acute angle is an angle that measures less than $90^\circ$.

If the angles are $\angle A$, $\angle B$, and $\angle C$, then for an acute-angled triangle, $\angle A < 90^\circ$, $\angle B < 90^\circ$, and $\angle C < 90^\circ$.

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2. Right-angled Triangle (or Right Triangle):

A right-angled triangle is a triangle in which one of the interior angles is a right angle. A right angle is an angle that measures exactly $90^\circ$.

If the angles are $\angle A$, $\angle B$, and $\angle C$, then for a right-angled triangle, one of the angles must be equal to $90^\circ$ (e.g., $\angle A = 90^\circ$). The other two angles in a right triangle must be acute.

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3. Obtuse-angled Triangle (or Obtuse Triangle):

An obtuse-angled triangle is a triangle in which one of the interior angles is an obtuse angle. An obtuse angle is an angle that measures more than $90^\circ$ but less than $180^\circ$.

If the angles are $\angle A$, $\angle B$, and $\angle C$, then for an obtuse-angled triangle, one of the angles must be obtuse (e.g., $90^\circ < \angle A < 180^\circ$). The other two angles in an obtuse triangle must be acute.

No, a triangle cannot have two obtuse angles.

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Justification:

We know that the sum of the interior angles in any triangle is always $180^\circ$.

An obtuse angle is defined as an angle that measures greater than $90^\circ$.

Let's assume, for the sake of contradiction, that a triangle has two obtuse angles. Let these angles be $\angle A$ and $\angle B$.

According to the definition of an obtuse angle:

If we add these two inequalities, we get:

$\angle A + \angle B > 90^\circ + 90^\circ$

$\angle A + \angle B > 180^\circ$

Now consider the sum of all three angles of the triangle:

$\angle A + \angle B + \angle C$

Since $\angle A + \angle B > 180^\circ$ and $\angle C$ must be a positive angle (as it's an angle in a triangle, $\angle C > 0^\circ$), their sum must be greater than $180^\circ$:

$\angle A + \angle B + \angle C > 180^\circ + \angle C > 180^\circ$

This result, $\angle A + \angle B + \angle C > 180^\circ$, contradicts the fundamental property that the sum of the interior angles of any triangle is exactly $180^\circ$ (equation i).

Therefore, our initial assumption that a triangle can have two obtuse angles must be false.

Hence, a triangle cannot have two obtuse angles.

No, a triangle cannot have two right angles.

---

Justification:

We know that the sum of the interior angles in any triangle is always $180^\circ$.

A right angle is defined as an angle that measures exactly $90^\circ$.

Let's assume, for the sake of contradiction, that a triangle has two right angles. Let these angles be $\angle A$ and $\angle B$.

According to the definition of a right angle:

If we add these two angle measures, we get:

$\angle A + \angle B = 90^\circ + 90^\circ$

$\angle A + \angle B = 180^\circ$

Now, let's substitute this sum into the triangle angle sum property (equation i):

$180^\circ + \angle C = 180^\circ$

Subtracting $180^\circ$ from both sides gives:

$\angle C = 180^\circ - 180^\circ$

$\angle C = 0^\circ$

However, an angle in a triangle must be a positive value, i.e., greater than $0^\circ$. An angle of $0^\circ$ means there is no angle, which would not form a triangle.

This result, $\angle C = 0^\circ$, contradicts the requirement that the third angle of a triangle must be greater than $0^\circ$.

Therefore, our initial assumption that a triangle can have two right angles must be false.

Hence, a triangle cannot have two right angles.

Given:

In $\triangle ABC$:

---

To Find:

The measure of $\angle C$.

---

Solution:

We use the Angle Sum Property of a Triangle, which states that the sum of the interior angles of any triangle is always $180^\circ$.

For $\triangle ABC$, this property is:

Substitute the given values of $\angle A$ and $\angle B$ into the equation:

$60^\circ + 70^\circ + \angle C = 180^\circ$

Combine the known angle measures:

$130^\circ + \angle C = 180^\circ$

To find $\angle C$, subtract $130^\circ$ from both sides of the equation:

$\angle C = 180^\circ - 130^\circ$

$\angle C = 50^\circ$

The measure of $\angle C$ is $50^\circ$.

The property used is the Angle Sum Property of a Triangle.

Given:

Exterior angle of a triangle = $110^\circ$.

One interior opposite angle = $50^\circ$.

---

To Find:

The measure of the other interior opposite angle.

---

Solution:

We use the Exterior Angle Property of a Triangle, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

Let the exterior angle be $E$ and the two interior opposite angles be $\angle A$ and $\angle B$. The property is:

Given $E = 110^\circ$ and let one interior opposite angle, say $\angle A = 50^\circ$. We need to find $\angle B$ (the other interior opposite angle).

Substitute the given values into the equation:

$110^\circ = 50^\circ + \angle B$

To find $\angle B$, subtract $50^\circ$ from both sides of the equation:

$\angle B = 110^\circ - 50^\circ$

$\angle B = 60^\circ$

The measure of the other interior opposite angle is $60^\circ$.

Given:

In an isosceles triangle:

One base angle = $40^\circ$.

---

To Find:

The measures of the vertex angle and the other base angle.

---

Solution:

In an isosceles triangle, the angles opposite the equal sides are equal. These are called the base angles.

Since one base angle is given as $40^\circ$, the other base angle must also be equal to this measure.

Let the base angles be $\angle B_1$ and $\angle B_2$, and the vertex angle be $\angle V$.

The sum of the interior angles in any triangle is $180^\circ$ (Angle Sum Property).

Substitute the values of the base angles:

$\angle V + 40^\circ + 40^\circ = 180^\circ$

Combine the known angle measures:

$\angle V + 80^\circ = 180^\circ$

To find the vertex angle $\angle V$, subtract $80^\circ$ from both sides:

$\angle V = 180^\circ - 80^\circ$

$\angle V = 100^\circ$

Thus, the measure of the other base angle is $40^\circ$ and the measure of the vertex angle is $100^\circ$.

Given:

The angles of a triangle are in the ratio $1:2:3$.

---

To Find:

The measure of each angle.

Classification of the triangle based on its angles and sides.

---

Solution:

Let the angles of the triangle be represented as $1x$, $2x$, and $3x$, based on the given ratio.

We know that the sum of the interior angles in any triangle is $180^\circ$ (Angle Sum Property of a Triangle).

So, we can set up the equation:

Combine the terms on the left side:

$6x = 180^\circ$

To find the value of $x$, divide both sides by 6:

$x = \frac{180^\circ}{6}$

$x = 30^\circ$

Now, substitute the value of $x$ to find the measure of each angle:

First angle = $1x = 1 \times 30^\circ = 30^\circ$

Second angle = $2x = 2 \times 30^\circ = 60^\circ$

Third angle = $3x = 3 \times 30^\circ = 90^\circ$

The measures of the angles of the triangle are $30^\circ$, $60^\circ$, and $90^\circ$.

---

Classification based on Angles:

The angles are $30^\circ$, $60^\circ$, and $90^\circ$. Since one of the angles is exactly $90^\circ$, the triangle is a Right-angled Triangle.

---

Classification based on Sides:

The angles are $30^\circ$, $60^\circ$, and $90^\circ$. All three angles are different ($30^\circ \neq 60^\circ \neq 90^\circ$).

In a triangle, sides opposite to equal angles are equal. Conversely, if all angles are different, then all sides opposite to these angles must also be different lengths.

Since all three angles are unequal, all three sides of the triangle must be unequal in length.

Therefore, based on its side lengths, the triangle is a Scalene Triangle.

Definition of a Median:

A median of a triangle is a line segment joining a vertex of the triangle to the midpoint of the side opposite to that vertex.

Every triangle has exactly three medians, one from each vertex.

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Drawing the Median from Vertex P to Side QR in $\triangle PQR$:

To draw the median from vertex P to side QR in $\triangle PQR$, follow these steps:

1. Draw a triangle and label its vertices as P, Q, and R.

2. Identify the side opposite to vertex P. This is side QR.

3. Find the midpoint of the side QR. Let's call this midpoint M.

4. Draw a line segment connecting vertex P to the midpoint M on side QR.

The line segment PM is the median from vertex P to side QR.

When drawing, make sure that M is exactly in the middle of the segment QR, so that QM = MR.

(Note: A visual diagram would typically accompany this description, showing $\triangle PQR$ with point M on QR such that QM=MR, and a line segment PM.)

Definition of an Altitude:

An altitude of a triangle is a perpendicular line segment drawn from a vertex of the triangle to the opposite side (or the extension of the opposite side). The point where the altitude intersects the opposite side (or its extension) is called the foot of the altitude.

Every triangle has exactly three altitudes, one from each vertex.

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Drawing the Altitude from Vertex X to Side YZ in $\triangle XYZ$:

To draw the altitude from vertex X to side YZ in $\triangle XYZ$, follow these steps:

1. Draw a triangle and label its vertices as X, Y, and Z.

2. Identify the side opposite to vertex X. This is side YZ.

3. From vertex X, draw a line segment that is perpendicular to the side YZ. This line segment should meet the side YZ (or its extension) at a $90^\circ$ angle.

4. Let the point where this perpendicular segment meets the line containing YZ be denoted by F. The line segment XF is the altitude from vertex X to side YZ.

(Note: The foot of the altitude F may lie on the segment YZ (for acute or right triangles) or outside the segment YZ on the extension of the line YZ (for obtuse triangles). The drawing should accurately reflect the case.)

When drawing, use a protractor or set square to ensure the angle at F is exactly $90^\circ$.

(Note: A visual diagram would typically accompany this description, showing $\triangle XYZ$ and the perpendicular line segment XF from X to YZ, with a right-angle symbol at F.)

Triangle Inequality Theorem:

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

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If a triangle has sides with lengths $a$, $b$, and $c$, then the theorem can be expressed as three inequalities:

$a + b > c$

$a + c > b$

$b + c > a$

For three given line segments to form a triangle, all three of these inequalities must be true. If even one inequality is not satisfied, the segments cannot form a triangle.

No, a triangle cannot be formed with side lengths $3$ cm, $4$ cm, and $8$ cm.

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Justification using the Triangle Inequality Theorem:

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the given side lengths be $a = 3$ cm, $b = 4$ cm, and $c = 8$ cm.

We need to check if the following three inequalities are true:

1. $a + b > c$

2. $a + c > b$

3. $b + c > a$

Let's check each inequality:

1. $3 + 4 > 8$

$7 > 8$

This statement is False.

2. $3 + 8 > 4$

$11 > 4$

This statement is True.

3. $4 + 8 > 3$

$12 > 3$

This statement is True.

For the three segments to form a triangle, all three inequalities must be true. Since the first inequality ($3 + 4 > 8$) is false, the given side lengths 3 cm, 4 cm, and 8 cm cannot form a triangle.

Pythagoras Property (or Pythagorean Theorem):

The Pythagoras property states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

---

If a right-angled triangle has legs of lengths $a$ and $b$, and the hypotenuse of length $c$, the property can be written as:

$a^2 + b^2 = c^2$

This property is fundamental for right-angled triangles.

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In a right-angled triangle ABC, right-angled at B:

The right angle is at vertex B.

The side opposite to the right angle (at vertex B) is the side connecting vertices A and C.

Therefore, in $\triangle ABC$ right-angled at B, the hypotenuse is the side AC.

Given:

A right-angled triangle with lengths of the two legs as 6 cm and 8 cm.

Let the lengths of the legs be $a = 6$ cm and $b = 8$ cm.

---

To Find:

The length of the hypotenuse, let it be $c$.

---

Solution:

We use the Pythagoras Property, which applies to right-angled triangles:

The square of the hypotenuse is equal to the sum of the squares of the other two sides.

In terms of $a$, $b$, and $c$ (where $c$ is the hypotenuse):

$c^2 = a^2 + b^2$

Substitute the given values for $a$ and $b$:

$c^2 = (6 \text{ cm})^2 + (8 \text{ cm})^2$

$c^2 = 36 \text{ cm}^2 + 64 \text{ cm}^2$

$c^2 = 100 \text{ cm}^2$

To find the length of the hypotenuse $c$, take the square root of both sides:

$c = \sqrt{100 \text{ cm}^2}$

$c = 10$ cm

(Since length must be positive)

The length of the hypotenuse is 10 cm.

Given:

A right-angled triangle.

Length of the hypotenuse, $c = 13$ cm.

Length of one other side (a leg), let's say $a = 5$ cm.

---

To Find:

The length of the third side (the other leg), let's say $b$.

---

Solution:

We use the Pythagoras Property, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

The formula is:

$a^2 + b^2 = c^2$

Substitute the given values for $a$ and $c$ into the equation:

$(5 \text{ cm})^2 + b^2 = (13 \text{ cm})^2$

$25 \text{ cm}^2 + b^2 = 169 \text{ cm}^2$

To find $b^2$, subtract $25 \text{ cm}^2$ from both sides:

$b^2 = 169 \text{ cm}^2 - 25 \text{ cm}^2$

$b^2 = 144 \text{ cm}^2$

To find the length of side $b$, take the square root of both sides:

$b = \sqrt{144 \text{ cm}^2}$

$b = 12$ cm

(Since length must be positive)

The length of the third side is 12 cm.

Given:

Side lengths: 9, 12, and 15.

---

To Check:

If these side lengths can form a right-angled triangle.

---

Solution:

We use the Converse of the Pythagoras Property. This property states that if, in a triangle, the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.

Let the side lengths be $a = 9$, $b = 12$, and $c = 15$. The longest side is 15, which would be the potential hypotenuse.

We need to check if $c^2 = a^2 + b^2$.

Calculate the square of the longest side:

Potential hypotenuse squared = $15^2 = 15 \times 15 = 225$

Calculate the sum of the squares of the other two sides:

Sum of squares of other sides = $9^2 + 12^2$

$9^2 = 9 \times 9 = 81$

$12^2 = 12 \times 12 = 144$

$9^2 + 12^2 = 81 + 144$

$\begin{array}{cc} & 8 & 1 \\ + & 1 & 4 & 4 \\ \hline & 2 & 2 & 5 \\ \hline \end{array}$

Sum of squares of other sides = 225

Compare the results:

Since the square of the longest side is equal to the sum of the squares of the other two sides ($15^2 = 9^2 + 12^2$), according to the converse of the Pythagoras property, a triangle with these side lengths is a right-angled triangle.

Therefore, yes, 9, 12, and 15 are the sides of a right-angled triangle.

Interior Angles of an Equilateral Triangle:

In an equilateral triangle, all three sides are equal in length. A property of triangles states that angles opposite to equal sides are equal. Since all three sides are equal, all three interior angles of an equilateral triangle must be equal in measure.

Let the measure of each interior angle be $x$. The sum of the interior angles in any triangle is $180^\circ$ (Angle Sum Property).

So, for an equilateral triangle:

$x + x + x = 180^\circ$

$3x = 180^\circ$

To find $x$, divide both sides by 3:

$x = \frac{180^\circ}{3}$

$x = 60^\circ$

So, the measure of each interior angle in an equilateral triangle is $60^\circ$.

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Exterior Angles of an Equilateral Triangle:

An exterior angle of a triangle and its adjacent interior angle form a linear pair, meaning they lie on a straight line and their measures sum up to $180^\circ$.

Let an interior angle be $\angle I$ and its adjacent exterior angle be $\angle E$.

For an equilateral triangle, each interior angle is $60^\circ$. So, $\angle I = 60^\circ$.

Substitute the value of the interior angle:

$60^\circ + \angle E = 180^\circ$

To find $\angle E$, subtract $60^\circ$ from both sides:

$\angle E = 180^\circ - 60^\circ$

$\angle E = 120^\circ$

Since all interior angles are equal, all exterior angles will also be equal. The measure of each exterior angle in an equilateral triangle is $120^\circ$.

To determine if a triangle can have angles $30^\circ$, $40^\circ$, and $110^\circ$, we use the Angle Sum Property of a Triangle.

---

Justification:

The Angle Sum Property states that the sum of the interior angles of any triangle is always $180^\circ$.

Let the given angles be $\angle A = 30^\circ$, $\angle B = 40^\circ$, and $\angle C = 110^\circ$.

We calculate the sum of these angles:

Sum $= 30^\circ + 40^\circ + 110^\circ$

Sum $= 70^\circ + 110^\circ$

Sum $= 180^\circ$

The sum of the given angles is exactly $180^\circ$.

Since the sum of the interior angles is $180^\circ$, these angle measures can form a triangle.

Therefore, yes, a triangle can have angles measuring $30^\circ$, $40^\circ$, and $110^\circ$.

This specific triangle would be classified as an obtuse-angled triangle because one of its angles ($110^\circ$) is obtuse (greater than $90^\circ$).

Given:

In a triangle, let the two given angles be $\angle A = 55^\circ$ and $\angle B = 65^\circ$.

---

To Find:

The measure of the third angle ($\angle C$).

Classification of the triangle based on angles.

---

Solution:

We use the Angle Sum Property of a Triangle, which states that the sum of the interior angles of any triangle is always $180^\circ$.

For this triangle, the property is:

Substitute the given values of $\angle A$ and $\angle B$ into the equation:

$55^\circ + 65^\circ + \angle C = 180^\circ$

Combine the known angle measures:

$120^\circ + \angle C = 180^\circ$

To find $\angle C$, subtract $120^\circ$ from both sides of the equation:

$\angle C = 180^\circ - 120^\circ$

$\angle C = 60^\circ$

The measure of the third angle is $60^\circ$.

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Classification based on Angles:

The measures of the three angles of the triangle are $55^\circ$, $65^\circ$, and $60^\circ$.

We examine each angle to classify the triangle:

$55^\circ < 90^\circ$ (Acute angle)

$65^\circ < 90^\circ$ (Acute angle)

$60^\circ < 90^\circ$ (Acute angle)

Since all three interior angles are acute angles (less than $90^\circ$), the triangle is an Acute-angled Triangle.

Given:

An exterior angle of a triangle = $100^\circ$.

One interior opposite angle = $45^\circ$.

---

To Find:

The measures of all interior angles of the triangle.

---

Solution:

Let the exterior angle be $E = 100^\circ$.

Let the two interior opposite angles be $\angle A$ and $\angle B$. We are given one of them, say $\angle A = 45^\circ$.

Let the third interior angle, which is adjacent to the exterior angle $E$, be $\angle C$.

We use the Exterior Angle Property of a Triangle, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

Substitute the given values $E = 100^\circ$ and $\angle A = 45^\circ$:

$100^\circ = 45^\circ + \angle B$

To find $\angle B$, subtract $45^\circ$ from both sides:

$\angle B = 100^\circ - 45^\circ$

$\angle B = 55^\circ$

So, one of the interior angles is $45^\circ$, and the other interior opposite angle is $55^\circ$.

Now we need to find the third interior angle, $\angle C$. The exterior angle $E$ and the adjacent interior angle $\angle C$ form a linear pair, so their sum is $180^\circ$.

To find $\angle C$, subtract $100^\circ$ from both sides:

$\angle C = 180^\circ - 100^\circ$

$\angle C = 80^\circ$

Alternatively, using the Angle Sum Property of a Triangle, the sum of the three interior angles ($\angle A$, $\angle B$, and $\angle C$) is $180^\circ$:

Substitute the values we found for $\angle A$ and $\angle B$:

$45^\circ + 55^\circ + \angle C = 180^\circ$

$100^\circ + \angle C = 180^\circ$

$\angle C = 180^\circ - 100^\circ$

$\angle C = 80^\circ$

Both methods give the same result for the third angle.

The measures of the three interior angles of the triangle are $45^\circ$, $55^\circ$, and $80^\circ$.

Given:

In an isosceles triangle:

---

To Find:

The measure of each base angle.

---

Solution:

In an isosceles triangle, the two base angles opposite the equal sides are equal in measure.

Let the measure of each base angle be $x$.

The sum of the interior angles in any triangle is $180^\circ$ (Angle Sum Property of a Triangle).

So, we can set up the equation:

Combine the terms involving $x$:

$80^\circ + 2x = 180^\circ$

To isolate the term with $x$, subtract $80^\circ$ from both sides:

$2x = 180^\circ - 80^\circ$

$2x = 100^\circ$

To find the value of $x$, divide both sides by 2:

$x = \frac{100^\circ}{2}$

$x = 50^\circ$

The measure of each base angle is $50^\circ$.

Given:

Two sides of a triangle have lengths $5$ cm and $12$ cm.

Let the lengths of these sides be $a = 5$ cm and $b = 12$ cm.

---

To Find:

The range of possible lengths for the third side.

---

Solution:

We use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the length of the third side be $c$ cm.

According to the Triangle Inequality Theorem, the following three inequalities must be true:

1. The sum of the two given sides must be greater than the third side:

$a + b > c$

$5 + 12 > c$

2. The sum of the first given side and the third side must be greater than the second given side:

$a + c > b$

$5 + c > 12$

Subtracting 5 from both sides:

3. The sum of the second given side and the third side must be greater than the first given side:

$b + c > a$

$12 + c > 5$

Subtracting 12 from both sides:

Since the length of a side must be a positive value, $c$ must be greater than 0. Inequality (iii) $c > -7$ is always satisfied if $c > 0$. The important conditions are $c < 17$ (from i) and $c > 7$ (from ii).

Combining inequalities (i) and (ii), we get:

$7 < c < 17$

This means the length of the third side must be greater than 7 cm and less than 17 cm.

The length of the third side must lie between 7 cm and 17 cm.

Angle Sum Property of a Triangle:

The Angle Sum Property of a triangle states that the sum of the measures of the three interior angles of any triangle is always $180^\circ$.

---

Proof that the sum of interior angles of a triangle is $180^\circ$:

Consider a triangle $\triangle ABC$. Let its interior angles be $\angle 1$, $\angle 2$, and $\angle 3$ at vertices A, B, and C respectively.

(Imagine a triangle ABC with angles $\angle 1$ at A, $\angle 2$ at B, and $\angle 3$ at C)

Construction:

Draw a line $XY$ passing through vertex A such that $XY$ is parallel to the side BC.

(Imagine the line XY above triangle ABC, passing through A, with X on the left and Y on the right, parallel to BC.)

Let the angle formed by line $XY$ and side AB to the left of A be $\angle 4$.

Let the angle formed by line $XY$ and side AC to the right of A be $\angle 5$.

(Imagine $\angle 4$ between XA and AB, and $\angle 5$ between AY and AC. $\angle 1$ is between AB and AC)

Proof Steps:

Since line $XY$ is a straight line, the angles on a straight line sum up to $180^\circ$. The angles $\angle 4$, $\angle 1$, and $\angle 5$ lie on the straight line XY at point A.

Now, consider the parallel lines $XY$ and $BC$, and the transversal AB.

The angles $\angle 4$ and $\angle 2$ (angle at B) are alternate interior angles.

According to the property of parallel lines, alternate interior angles are equal.

Next, consider the parallel lines $XY$ and $BC$, and the transversal AC.

The angles $\angle 5$ and $\angle 3$ (angle at C) are alternate interior angles.

According to the property of parallel lines, alternate interior angles are equal.

Now, substitute the values of $\angle 4$ and $\angle 5$ from the alternate interior angle properties into the equation for angles on the straight line:

Substitute $\angle 4 = \angle 2$ and $\angle 5 = \angle 3$:

Rearranging the terms, we get the sum of the interior angles of $\triangle ABC$:

This shows that the sum of the interior angles of any triangle is $180^\circ$.

Relationship between an Exterior Angle and Interior Opposite Angles:

An exterior angle of a triangle is formed when one side of the triangle is extended. It forms a linear pair with the adjacent interior angle.

The interior opposite angles (also called remote interior angles) are the two interior angles of the triangle that are not adjacent to the exterior angle.

The relationship is given by the Exterior Angle Property of a Triangle:

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.

---

Given:

In $\triangle ABC$, the exterior angle at vertex C is $120^\circ$.

One interior opposite angle, $\angle A = 55^\circ$.

---

To Find:

The measures of $\angle B$ and $\angle C$ (interior angles).

---

Solution:

Let the exterior angle at C be $\angle ACD = 120^\circ$, where D is a point on the extension of BC beyond C.

The interior opposite angles to $\angle ACD$ are $\angle A$ and $\angle B$. The adjacent interior angle is $\angle C$ (or $\angle ACB$).

Using the Exterior Angle Property:

Substitute the given values:

$120^\circ = 55^\circ + \angle B$

To find $\angle B$, subtract $55^\circ$ from both sides:

$120^\circ - 55^\circ = \angle B$

Now, to find the interior angle $\angle C$, we know that the exterior angle $\angle ACD$ and the adjacent interior angle $\angle C$ form a linear pair. Their sum is $180^\circ$.

To find $\angle C$, subtract $120^\circ$ from both sides:

$\angle C = 180^\circ - 120^\circ$

Alternatively, we can find $\angle C$ using the Angle Sum Property after finding $\angle B$.

Substitute the values $\angle A = 55^\circ$ and $\angle B = 65^\circ$:

$55^\circ + 65^\circ + \angle C = 180^\circ$

$120^\circ + \angle C = 180^\circ$

$\angle C = 180^\circ - 120^\circ$

$\angle C = 60^\circ$

The measure of angle $\angle B$ is $65^\circ$, and the measure of angle $\angle C$ is $60^\circ$.

Given:

Length of the ladder (Hypotenuse) = $15$ meters.

Height of the window from the ground (One leg of the right triangle) = $9$ meters.

The wall is perpendicular to the ground, so the ladder, the wall, and the ground form a right-angled triangle.

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To Find:

The distance of the foot of the ladder from the wall (The other leg of the right triangle).

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Solution:

Let the distance of the foot of the ladder from the wall be $d$ meters.

In the right-angled triangle formed, the ladder is the hypotenuse, and the wall height and the distance from the foot of the ladder to the wall are the two legs.

According to the Pythagoras Property:

(Hypotenuse)$^2$ = (Leg 1)$^2$ + (Leg 2)$^2$

Substituting the given values:

Calculate the squares:

$15^2 = 15 \times 15 = 225$

$9^2 = 9 \times 9 = 81$

Substitute these values back into the equation:

$225 = 81 + d^2$

To find $d^2$, subtract 81 from both sides of the equation:

$d^2 = 225 - 81$

$\begin{array}{cc} & 2 & 2 & 5 \\ - & & 8 & 1 \\ \hline & 1 & 4 & 4 \\ \hline \end{array}$

$d^2 = 144$

To find $d$, take the square root of $144$:

$d = \sqrt{144}$

$d = 12$

(Since distance must be positive)

The distance of the foot of the ladder from the wall is $12$ meters.

Definitions:

A median of a triangle is a line segment from a vertex to the midpoint of the opposite side.

An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or its extension).

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Drawing for $\triangle PQR$ (right-angled at Q):

1. Draw a right-angled triangle and label the vertices P, Q, and R, ensuring that the angle at Q is $90^\circ$. Side PR is the hypotenuse.

2. Median from Q to PR: Find the midpoint of the hypotenuse PR. Let's call this point M. Draw the line segment QM. This is the median from Q to PR.

3. Altitude from Q to PR: Draw a line segment from Q that is perpendicular to the hypotenuse PR. Let the point where this perpendicular meets PR be F. Draw the line segment QF. This is the altitude from Q to PR.

(Note: A visual diagram showing $\triangle PQR$ with $\angle Q = 90^\circ$, the median QM where M is the midpoint of PR, and the altitude QF where $\angle QFP = 90^\circ$ or $\angle QFR = 90^\circ$ would be included here.)

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Discussion on whether Median and Altitude from Q are the same:

In a general right-angled triangle, the median from the right-angle vertex and the altitude from the right-angle vertex are not the same.

The median QM connects Q to the midpoint M of PR. The altitude QF is perpendicular to PR at point F.

Unless the point F (foot of the altitude) coincides with the point M (midpoint of the hypotenuse), the segments QM and QF will be different lengths and directions.

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When they might be the same:

The median and the altitude from the right-angle vertex (Q) are the same if and only if the right-angled triangle is also an isosceles triangle, with the two legs being equal in length ($QP = QR$).

In an isosceles right-angled triangle, the altitude from the vertex containing the right angle to the hypotenuse bisects the hypotenuse. This means the foot of the altitude (F) falls exactly on the midpoint (M) of the hypotenuse. In this specific case, the altitude QF and the median QM are the same line segment.

So, they are the same only when $\triangle PQR$ is an isosceles right-angled triangle where $QP = QR$.

Given:

The angles of a triangle are $(2x)^\circ$, $(3x-5)^\circ$, and $(x+15)^\circ$.

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To Find:

The value of $x$.

The measure of each angle.

Classification of the triangle based on its angles.

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Solution:

We use the Angle Sum Property of a Triangle, which states that the sum of the interior angles of any triangle is always $180^\circ$.

So, the sum of the given angles must be $180^\circ$:

Now, we solve for $x$:

$2x + 3x - 5 + x + 15 = 180$

Combine the terms with $x$ and the constant terms:

$(2x + 3x + x) + (-5 + 15) = 180$

$6x + 10 = 180$

Subtract 10 from both sides:

$6x = 180 - 10$

$6x = 170$

Divide both sides by 6:

$x = \frac{170}{6}$

$x = \frac{85}{3}$

The value of $x$ is $\frac{85}{3}$.

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Now, we find the measure of each angle by substituting $x = \frac{85}{3}$:

First angle: $2x = 2 \times \frac{85}{3} = \frac{170}{3}^\circ$

Second angle: $3x - 5 = 3 \times \frac{85}{3} - 5 = 85 - 5 = 80^\circ$

Third angle: $x + 15 = \frac{85}{3} + 15 = \frac{85}{3} + \frac{45}{3} = \frac{85 + 45}{3} = \frac{130}{3}^\circ$

The measures of the angles are $\frac{170}{3}^\circ$, $80^\circ$, and $\frac{130}{3}^\circ$.

Let's verify the sum: $\frac{170}{3} + 80 + \frac{130}{3} = \frac{170}{3} + \frac{240}{3} + \frac{130}{3} = \frac{170 + 240 + 130}{3} = \frac{540}{3} = 180^\circ$. The sum is correct.

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Classification based on Angles:

The angle measures are $\frac{170}{3}^\circ \approx 56.67^\circ$, $80^\circ$, and $\frac{130}{3}^\circ \approx 43.33^\circ$.

We check if any angle is $90^\circ$ or greater than $90^\circ$:

$\frac{170}{3} < \frac{270}{3} = 90^\circ$

$80^\circ < 90^\circ$

$\frac{130}{3} < \frac{270}{3} = 90^\circ$

Since all three angles are less than $90^\circ$, they are all acute angles.

Therefore, the triangle is an Acute-angled Triangle.

We will use the Triangle Inequality Theorem to check if the given side lengths can form a triangle, and the Converse of the Pythagoras Property to check if a triangle (if formed) is a right-angled triangle.

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(a) Side lengths: 5 cm, 12 cm, 13 cm

Let $a = 5$, $b = 12$, $c = 13$.

Check Triangle Inequality:

1. $a + b > c \implies 5 + 12 > 13 \implies 17 > 13$ (True)

2. $a + c > b \implies 5 + 13 > 12 \implies 18 > 12$ (True)

3. $b + c > a \implies 12 + 13 > 5 \implies 25 > 5$ (True)

Since all three inequalities are true, these side lengths can form a triangle.

Check for Right-angled Triangle (using Converse of Pythagoras Property):

The longest side is 13 (potential hypotenuse). We check if the square of the longest side is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$).

$a^2 = 5^2 = 25$

$b^2 = 12^2 = 144$

$c^2 = 13^2 = 169$

Is $a^2 + b^2 = c^2$? $25 + 144 = 169$.

$169 = 169$ (True)

Since $5^2 + 12^2 = 13^2$, the triangle is a right-angled triangle.

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(b) Side lengths: 4 cm, 6 cm, 10 cm

Let $a = 4$, $b = 6$, $c = 10$.

Check Triangle Inequality:

1. $a + b > c \implies 4 + 6 > 10 \implies 10 > 10$ (False)

Since the first inequality is false, we do not need to check the others. These side lengths cannot form a triangle.

Because they cannot form a triangle, it is not a right-angled triangle.

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(c) Side lengths: 7 cm, 8 cm, 11 cm

Let $a = 7$, $b = 8$, $c = 11$.

Check Triangle Inequality:

1. $a + b > c \implies 7 + 8 > 11 \implies 15 > 11$ (True)

2. $a + c > b \implies 7 + 11 > 8 \implies 18 > 8$ (True)

3. $b + c > a \implies 8 + 11 > 7 \implies 19 > 7$ (True)

Since all three inequalities are true, these side lengths can form a triangle.

Check for Right-angled Triangle (using Converse of Pythagoras Property):

The longest side is 11 (potential hypotenuse). We check if the square of the longest side is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$).

$a^2 = 7^2 = 49$

$b^2 = 8^2 = 64$

$c^2 = 11^2 = 121$

Is $a^2 + b^2 = c^2$? $49 + 64 = 113$.

$113 = 121$ (False)

Since $7^2 + 8^2 \neq 11^2$, the triangle is not a right-angled triangle.

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Summary:

(a) 5 cm, 12 cm, 13 cm: Can form a triangle and is a right-angled triangle.

(b) 4 cm, 6 cm, 10 cm: Cannot form a triangle.

(c) 7 cm, 8 cm, 11 cm: Can form a triangle but is not a right-angled triangle.

Given:

In $\triangle ABC$, AD is the median to side BC.

AD is extended to a point E such that AD = DE.

(Imagine $\triangle ABC$, D is the midpoint of BC. Line segment AD is extended straight through D to point E such that the length of AD is equal to the length of DE. Then point E is connected to C).

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To Prove (Informally):

$\triangle ABD \cong \triangle ECD$

To find the relationship between sides AB and CE.

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Informal Proof of Congruence:

Consider $\triangle ABD$ and $\triangle ECD$.

1. Side AD = Side DE

(This is given by the construction).

2. Side BD = Side CD

($\text{AD}$ is a median of $\triangle ABC$ to side $\text{BC}$. By definition, a median connects a vertex to the midpoint of the opposite side. Therefore, D is the midpoint of BC, which means BD = CD).

3. Angle $\angle ADB$ = Angle $\angle EDC$

(These two angles are vertically opposite angles formed by the intersection of lines AE and BC at point D. Vertically opposite angles are always equal).

We have shown that two sides (AD and BD) and the included angle ($\angle ADB$) of $\triangle ABD$ are equal to two sides (DE and CD) and the included angle ($\angle EDC$) of $\triangle ECD$.

Therefore, by the SAS (Side-Angle-Side) Congruence Criterion, $\triangle ABD$ is congruent to $\triangle ECD$.

$\triangle ABD \cong \triangle ECD$ (SAS Congruence)

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Relationship between AB and CE:

Since $\triangle ABD \cong \triangle ECD$, their corresponding parts are equal (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

The side AB in $\triangle ABD$ corresponds to the side CE in $\triangle ECD$.

Therefore, we can say that side AB is equal in length to side CE.

$\text{AB = CE}$ (CPCTC)

Given:

Height of the electric pole = $10$ meters.

Distance of the point on the ground from the base of the pole = $24$ meters.

The pole is vertical to the ground, forming a $90^\circ$ angle with the ground.

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To Find:

The length of the wire required.

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Solution:

The situation described forms a right-angled triangle. The electric pole is one leg, the distance on the ground from the base of the pole is the other leg, and the wire is the hypotenuse.

Let the height of the pole be $h = 10$ m.

Let the distance from the base of the pole to the point on the ground be $d = 24$ m.

Let the length of the wire be $w$.

According to the Pythagoras Property for a right-angled triangle:

In this specific problem:

Substitute the given values for $h$ and $d$:

$w^2 = (10 \text{ m})^2 + (24 \text{ m})^2$

$w^2 = 100 \text{ m}^2 + 576 \text{ m}^2$

$w^2 = 676 \text{ m}^2$

To find the length of the wire $w$, take the square root of both sides of the equation:

$w = \sqrt{676 \text{ m}^2}$

$w = 26$ m

(We take the positive square root as length cannot be negative)

The length of the wire required is $26$ meters.

Given:

Perimeter of an isosceles triangle = $30$ cm.

Length of each equal side = $12$ cm.

Height corresponding to the third side = $6$ cm.

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To Find:

The length of the third side.

The area of the triangle.

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Solution - Finding the Third Side:

Let the lengths of the sides of the isosceles triangle be $a$, $b$, and $c$. In an isosceles triangle, two sides are equal. Let the equal sides be $a = 12$ cm and $b = 12$ cm. Let the third side be $c$.

The perimeter of a triangle is the sum of its side lengths:

Substitute the given values:

$30 = 12 + 12 + c$

Simplify the equation:

$30 = 24 + c$

Subtract 24 from both sides to find $c$:

$c = 30 - 24$

$c = 6$ cm

The length of the third side is $6$ cm.

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Solution - Finding the Area:

The area of a triangle can be calculated using the formula:

We found the length of the third side to be $6$ cm. The height corresponding to this side is given as $6$ cm.

Let the base be the third side, $b_3 = 6$ cm.

Let the height corresponding to this base be $h_3 = 6$ cm.

Substitute these values into the area formula:

Area = $\frac{1}{2} \times 6 \text{ cm} \times 6 \text{ cm}$

Calculate the area:

Area = $\frac{1}{2} \times 36 \text{ cm}^2$

Area = $18 \text{ cm}^2$

The area of the triangle is $18$ square centimeters.

Given:

A triangle with interior angles $50^\circ$, $x$, and $y$.

An exterior angle adjacent to the interior angle $x$ is $110^\circ$.

(Assume the interior angles are at vertices A, B, C with $\angle A = 50^\circ$, $\angle B = x$, $\angle C = y$, and the exterior angle is at B).

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To Find:

The measures of angles $x$ and $y$.

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Solution:

First, we find the measure of angle $x$. The interior angle $x$ and the exterior angle $110^\circ$ at the same vertex form a linear pair. The sum of angles in a linear pair is $180^\circ$.

Subtract $110^\circ$ from both sides to solve for $x$:

$x = 180^\circ - 110^\circ$

$x = 70^\circ$

The measure of angle $x$ is $70^\circ$.

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Next, we find the measure of angle $y$. We can use the Angle Sum Property of a Triangle, which states that the sum of the interior angles of any triangle is $180^\circ$. The interior angles are $50^\circ$, $x$, and $y$.

Substitute the value of $x = 70^\circ$ that we just found:

$50^\circ + 70^\circ + y = 180^\circ$

Combine the known angle measures:

$120^\circ + y = 180^\circ$

Subtract $120^\circ$ from both sides to solve for $y$:

$y = 180^\circ - 120^\circ$

$y = 60^\circ$

The measure of angle $y$ is $60^\circ$.

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Verification (using Exterior Angle Property):

The Exterior Angle Property states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. The exterior angle is $110^\circ$, and the two opposite interior angles are $50^\circ$ and $y$.

$y = 110^\circ - 50^\circ$

$y = 60^\circ$

This confirms our value for $y$.

The measures of the angles are $x = 70^\circ$ and $y = 60^\circ$.

The problem describes movement in two perpendicular directions (East and North). This situation forms a right-angled triangle where the path east and the path north are the two legs, and the distance from the starting point is the hypotenuse.

(A diagram should be drawn showing a point as the start, a horizontal line segment representing 8 km East, a vertical line segment upwards from the end of the first segment representing 15 km North, and a diagonal line segment connecting the start point to the end point, forming the hypotenuse of a right-angled triangle).

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Given:

Distance walked East = $8$ km (Let this be one leg, $a$).

Distance walked North = $15$ km (Let this be the other leg, $b$).

The path East and the path North are perpendicular, forming a right angle.

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To Find:

The distance from the starting point (the length of the hypotenuse, $c$).

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Solution:

We use the Pythagoras Property for a right-angled triangle, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Substitute the given values $a = 8$ km and $b = 15$ km:

$c^2 = (8 \text{ km})^2 + (15 \text{ km})^2$

Calculate the squares:

$8^2 = 8 \times 8 = 64$

$15^2 = 15 \times 15 = 225$

Substitute these values back into the equation:

$c^2 = 64 + 225$

Perform the addition:

$\begin{array}{cc} & & 6 & 4 \\ + & 2 & 2 & 5 \\ \hline & 2 & 8 & 9 \\ \hline \end{array}$

$c^2 = 289$

To find the length of the hypotenuse $c$, take the square root of $289$:

$c = \sqrt{289}$

$c = 17$

(Since distance must be positive, we take the positive square root)

The distance of the man from his starting point is $17$ km.

Given:

The sides of a triangle are in the ratio $3:4:5$.

The perimeter of the triangle is $60$ cm.

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To Find:

The length of each side.

The classification of the triangle based on its sides and angles.

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Solution - Finding Side Lengths:

Let the lengths of the sides of the triangle be $3x$, $4x$, and $5x$, based on the given ratio.

The perimeter of a triangle is the sum of its side lengths.

We are given that the perimeter is $60$ cm. So, we can set up the equation:

$3x + 4x + 5x = 60$

Combine the terms on the left side:

$12x = 60$

To find the value of $x$, divide both sides by 12:

$x = \frac{60}{12}$

$x = 5$

Now, substitute the value of $x$ to find the length of each side:

First side = $3x = 3 \times 5 = 15$ cm

Second side = $4x = 4 \times 5 = 20$ cm

Third side = $5x = 5 \times 5 = 25$ cm

The lengths of the sides are 15 cm, 20 cm, and 25 cm.

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Classification based on Sides:

The side lengths are 15 cm, 20 cm, and 25 cm. All three side lengths are different ($15 \neq 20 \neq 25$).

A triangle with all three sides of different lengths is called a Scalene Triangle.

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Classification based on Angles:

To classify the triangle based on angles, we can use the Converse of the Pythagoras Property. We check if the square of the longest side is equal to the sum of the squares of the other two sides.

The longest side is 25 (potential hypotenuse).

Calculate the square of the longest side: $25^2 = 25 \times 25 = 625$.

Calculate the sum of the squares of the other two sides: $15^2 + 20^2 = (15 \times 15) + (20 \times 20) = 225 + 400 = 625$.

Since $15^2 + 20^2 = 25^2$ (i.e., $225 + 400 = 625$), the condition of the converse of the Pythagoras property is satisfied.

Therefore, the triangle is a Right-angled Triangle.