Chapter 6 Lines And Angles (Additional Questions)

Explanation:

Let's examine the definitions of the given options:

A Ray is a part of a line that has one endpoint and extends infinitely in one direction.

A Line is a straight one-dimensional figure that has no thickness and extends endlessly in both directions.

A Line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its endpoints.

A Point is a location in space and has no dimension (length, width, or height).

The question describes "A part of a line with two endpoints". This definition precisely matches the definition of a line segment.

The correct option is (C) Line segment.

Explanation:

Let's define the different types of angles mentioned in the options:

An Acute angle is an angle whose measure is greater than $0^\circ$ and less than $90^\circ$.

An Obtuse angle is an angle whose measure is greater than $90^\circ$ and less than $180^\circ$.

A Right angle is an angle whose measure is exactly $90^\circ$.

A Reflex angle is an angle whose measure is greater than $180^\circ$ but less than $360^\circ$.

The question asks for an angle whose measure is more than $90^\circ$ but less than $180^\circ$. Comparing this condition with the definitions above, we see that this matches the definition of an obtuse angle.

The correct option is (B) Obtuse angle.

Explanation:

Two angles are called complementary angles if their sum is equal to $90^\circ$.

Let the measure of the first angle be $A_1$ and the measure of the second angle be $A_2$.

Given that the two angles are complementary, we have:

$A_1 + A_2 = 90^\circ$

We are given that one angle is $40^\circ$. Let's assume $A_1 = 40^\circ$.

Substituting this value into the equation:

$40^\circ + A_2 = 90^\circ$

To find the measure of the other angle ($A_2$), we subtract $40^\circ$ from both sides of the equation:

$A_2 = 90^\circ - 40^\circ$

$A_2 = 50^\circ$

Thus, the measure of the other angle is $50^\circ$.

Comparing this result with the given options, we find that the measure of the other angle is $50^\circ$, which corresponds to option (A).

The correct option is (A) $50^\circ$.

Explanation:

When two straight lines intersect each other, four angles are formed at the point of intersection.

Consider two lines, say line $AB$ and line $CD$, intersecting at point $O$. The angles formed are $\angle AOC$, $\angle COB$, $\angle BOD$, and $\angle DOA$.

Angles that are opposite to each other at the intersection point are called vertically opposite angles. In this case, $\angle AOC$ and $\angle BOD$ are a pair of vertically opposite angles, and $\angle COB$ and $\angle DOA$ are another pair of vertically opposite angles.

A fundamental geometric property states that when two lines intersect, the vertically opposite angles are equal in measure.

Therefore, $\angle AOC = \angle BOD$ and $\angle COB = \angle DOA$.

Based on this property, the vertically opposite angles formed when two straight lines intersect are always equal.

The correct option is (C) Equal.

Explanation:

We are given that lines $l$ and $m$ are parallel, and $t$ is a transversal intersecting these lines.

We are given the measure of $\angle 1$ as $70^\circ$.

We need to find the measure of $\angle 5$.

In the standard labeling of angles formed by a transversal intersecting two lines, $\angle 1$ and $\angle 5$ are a pair of corresponding angles.

One of the properties of parallel lines intersected by a transversal states that:

If two parallel lines are intersected by a transversal, then each pair of corresponding angles is equal.

Since line $l$ is parallel to line $m$ ($l \parallel m$) and $t$ is a transversal, the corresponding angles are equal in measure.

Therefore, the measure of $\angle 5$ is equal to the measure of $\angle 1$.

$\angle 5 = \angle 1$

Given that $\angle 1 = 70^\circ$, we have:

$\angle 5 = 70^\circ$

The measure of $\angle 5$ is $70^\circ$. This matches option (A).

The correct option is (A) $70^\circ$.

Explanation:

When a transversal line intersects two parallel lines, several pairs of angles are formed. The interior angles on the same side of the transversal are a pair of angles that are on the interior of the parallel lines and on the same side of the transversal.

Let lines $l$ and $m$ be parallel ($l \parallel m$), and let $t$ be a transversal intersecting $l$ and $m$. Let the angles formed be labeled.

The pairs of interior angles on the same side of the transversal are $\angle 3$ and $\angle 6$ (assuming standard numbering where $\angle 3$ and $\angle 6$ are between the parallel lines and on opposite sides of the transversal at the intersection points), and $\angle 4$ and $\angle 5$.

A property of parallel lines intersected by a transversal is that the sum of the interior angles on the same side of the transversal is always equal to $180^\circ$. These angles are supplementary.

So, $\angle 3 + \angle 6 = 180^\circ$ and $\angle 4 + \angle 5 = 180^\circ$.

The question asks for the sum of the interior angles on the same side of a transversal when two parallel lines are intersected.

Based on the property, this sum is $180^\circ$.

The correct option is (B) $180^\circ$.

Explanation:

When a transversal line intersects two other lines, alternate interior angles are pairs of angles that are on opposite sides of the transversal and between the two intersected lines.

Let lines $l$ and $m$ be intersected by a transversal $t$. Let's assume the standard labeling of angles from 1 to 8.

The pairs of alternate interior angles are $\angle 4$ and $\angle 5$, and $\angle 3$ and $\angle 6$.

There is a converse property related to parallel lines and transversals:

If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

This is one of the conditions used to determine if two lines are parallel.

The question states that the alternate interior angles are equal. According to the converse property, this implies that the lines must be parallel.

The correct option is (C) Parallel.

Explanation:

Consider any triangle, say triangle ABC. Let the interior angles be $\angle A$, $\angle B$, and $\angle C$.

A fundamental theorem in Euclidean geometry, known as the Angle Sum Property of a Triangle, states that the sum of the measures of the interior angles of any triangle is always equal to $180^\circ$.

Mathematically, for triangle ABC, we have:

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

This property holds true for all types of triangles (acute-angled, obtuse-angled, right-angled, equilateral, isosceles, scalene).

Therefore, the sum of the measures of all three interior angles in a triangle is $180^\circ$.

The correct option is (B) $180^\circ$.

Explanation:

Consider a triangle, say $\triangle ABC$. Let's extend one side, for example, side $BC$ to a point $D$. The angle $\angle ACD$ formed on the exterior of the triangle is called an exterior angle.

The interior angles of $\triangle ABC$ are $\angle ABC$ (or $\angle B$), $\angle BCA$ (or $\angle C$), and $\angle CAB$ (or $\angle A$).

The interior angle adjacent to the exterior angle $\angle ACD$ is $\angle BCA$ (or $\angle C$).

The two interior angles that are not adjacent to the exterior angle are $\angle ABC$ (or $\angle B$) and $\angle CAB$ (or $\angle A$). These are called the opposite interior angles to the exterior angle $\angle ACD$.

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

For the exterior angle $\angle ACD$, the property states:

$\angle ACD = \angle ABC + \angle CAB$

Based on the Exterior Angle Property, an exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The correct word to fill in the blank is "Opposite".

The correct option is (B) Opposite.

Explanation:

Adjacent angles are two angles that have a common vertex and a common side but no common interior points.

A linear pair is a pair of adjacent angles formed when two lines intersect. The non-common sides of the adjacent angles in a linear pair are opposite rays, forming a straight line.

The property of a linear pair is that the sum of the measures of the two angles that form a linear pair is always $180^\circ$. These angles are supplementary.

For example, if angles $\angle ABC$ and $\angle CBD$ form a linear pair, where $A$, $B$, and $D$ are collinear points, then:

$\angle ABC + \angle CBD = 180^\circ$

The question asks for the sum of two adjacent angles that form a linear pair.

Based on the definition and property of a linear pair, their sum is $180^\circ$.

The correct option is (B) $180^\circ$.

Explanation:

Let's analyze each option:

(A) Consecutive interior angles: These are interior angles on the same side of the transversal. They are supplementary (their sum is $180^\circ$) if and only if the two lines are parallel. They are not always equal when any two lines are intersected by a transversal.

(B) Adjacent angles: These angles share a common vertex and a common arm. Their measures vary depending on the specific intersection and do not have a general property of being equal when formed by a transversal intersecting two lines, unless under specific circumstances (e.g., bisection of an angle or forming a right angle). They are not always equal.

(C) Vertically opposite angles: When two lines intersect, the angles opposite to each other at the point of intersection are called vertically opposite angles. A fundamental geometric theorem states that vertically opposite angles are always equal.

(D) Angles in a linear pair: These are adjacent angles whose non-common sides form a straight line. Their sum is always $180^\circ$ (they are supplementary). They are equal only in the specific case where both angles are right angles ($90^\circ$). They are not always equal.

The question asks which pair of angles is always equal when two lines are intersected by a transversal. Vertically opposite angles fit this description because they are formed at each intersection point, and their equality is a property of intersecting lines themselves, regardless of whether the lines are parallel or not.

Therefore, the pair of angles that are always equal when two lines are intersected by a transversal is vertically opposite angles.

The correct option is (C) Vertically opposite angles.

Explanation:

We are given a triangle $\triangle ABC$ with the measures of two of its interior angles:

$\angle A = 50^\circ$

$\angle B = 60^\circ$

We need to find the measure of the third interior angle, $\angle C$.

According to the Angle Sum Property of a Triangle, the sum of the measures of the interior angles of any triangle is always $180^\circ$.

For $\triangle ABC$, this property can be written as:

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

Now, we substitute the given values of $\angle A$ and $\angle B$ into this equation:

$50^\circ + 60^\circ + \angle C = 180^\circ$

Combine the known angle measures:

$110^\circ + \angle C = 180^\circ$

To find $\angle C$, subtract $110^\circ$ from both sides of the equation:

$\angle C = 180^\circ - 110^\circ$

$\angle C = 70^\circ$

The measure of angle $\angle C$ is $70^\circ$. This matches option (A).

The correct option is (A) $70^\circ$.

Explanation:

Let's review the definitions of basic geometric concepts related to lines:

A Point is a location and has no dimensions.

A Line is a collection of points extending infinitely in both directions along a straight path. It has no endpoints.

A Line Segment is a part of a line that consists of two distinct endpoints and all the points between them.

A Ray is a part of a line that starts at a fixed point (called the endpoint or initial point) and extends infinitely in one direction. It has only one endpoint.

Comparing the definition of a ray with the given options:

(A) states it has one endpoint and extends infinitely in one direction. This matches the definition of a ray.

(B) describes a line segment.

(C) describes a line.

(D) is incorrect as a ray has one endpoint.

The correct option that describes a ray is the one stating it has one endpoint and extends infinitely in one direction.

The correct option is (A) One endpoint and extends infinitely in one direction.

Explanation:

Let's define the different types of angles based on their measures:

An Acute angle is an angle whose measure is greater than $0^\circ$ and less than $90^\circ$.

A Right angle is an angle whose measure is exactly $90^\circ$.

An Obtuse angle is an angle whose measure is greater than $90^\circ$ and less than $180^\circ$.

A Straight angle is an angle whose measure is exactly $180^\circ$.

A Reflex angle is an angle whose measure is greater than $180^\circ$ but less than $360^\circ$.

A Complete angle is an angle whose measure is exactly $360^\circ$.

Now, let's examine the measure of each angle given in the options:

(A) $75^\circ$: This measure is greater than $0^\circ$ and less than $90^\circ$. This is an acute angle.

(B) $150^\circ$: This measure is greater than $90^\circ$ and less than $180^\circ$. This is an obtuse angle.

(C) $210^\circ$: This measure is greater than $180^\circ$ and less than $360^\circ$. This is a reflex angle.

(D) $90^\circ$: This measure is exactly $90^\circ$. This is a right angle.

Based on the definitions, the angle with a measure of $210^\circ$ is a reflex angle.

The angle that is a reflex angle among the given options is $210^\circ$.

The correct option is (C) $210^\circ$.

Explanation:

Two lines are defined as parallel lines if they lie in the same plane and do not intersect, no matter how far they are extended in either direction.

A key property that distinguishes parallel lines is the distance between them.

The distance between two parallel lines is defined as the length of any perpendicular segment connecting a point on one line to the other line.

If the distance between the two lines were varying (increasing or decreasing), the lines would eventually either intersect (if the distance decreases to zero) or move farther apart indefinitely. Neither of these scenarios fits the definition of parallel lines.

Therefore, for two lines to be parallel, the perpendicular distance between them must remain the same at all points along the lines.

This means the distance between parallel lines is constant everywhere.

Based on the definition and property of parallel lines, the distance between them is constant everywhere.

The correct option is (B) Constant.

Explanation:

A linear pair is a pair of adjacent angles formed when two lines intersect. The sum of the measures of the angles in a linear pair is always $180^\circ$. In other words, angles forming a linear pair are supplementary.

Let the two angles forming a linear pair be $\alpha$ and $\beta$. Then, $\alpha + \beta = 180^\circ$.

Let's evaluate each statement:

(A) Two acute angles can form a linear pair.

An acute angle has a measure less than $90^\circ$. If two angles are acute, say $\alpha < 90^\circ$ and $\beta < 90^\circ$, then their sum is $\alpha + \beta < 90^\circ + 90^\circ = 180^\circ$. Since their sum is less than $180^\circ$, two acute angles cannot form a linear pair. This statement is False.

(B) Two obtuse angles can form a linear pair.

An obtuse angle has a measure greater than $90^\circ$ but less than $180^\circ$. If two angles are obtuse, say $90^\circ < \alpha < 180^\circ$ and $90^\circ < \beta < 180^\circ$, then their sum is $\alpha + \beta > 90^\circ + 90^\circ = 180^\circ$. Since their sum is greater than $180^\circ$, two obtuse angles cannot form a linear pair. This statement is False.

(C) An acute angle and an obtuse angle can form a linear pair.

Let one angle be acute, with measure $\alpha$ ($0^\circ < \alpha < 90^\circ$). Let the other angle be obtuse, with measure $\beta$ ($90^\circ < \beta < 180^\circ$). Can their sum be $180^\circ$? Yes, if they are supplementary. For example, if $\alpha = 30^\circ$ (acute), its supplement is $180^\circ - 30^\circ = 150^\circ$, which is an obtuse angle. If $\alpha$ is any acute angle ($0^\circ < \alpha < 90^\circ$), its supplement $180^\circ - \alpha$ will satisfy $180^\circ - 90^\circ < 180^\circ - \alpha < 180^\circ - 0^\circ$, which means $90^\circ < 180^\circ - \alpha < 180^\circ$. Thus, the supplement is an obtuse angle. Therefore, an acute angle and an obtuse angle can be supplementary and thus can form a linear pair. This statement is TRUE.

(D) Two right angles can form a linear pair.

A right angle has a measure of exactly $90^\circ$. If two angles are right angles, their sum is $90^\circ + 90^\circ = 180^\circ$. Since their sum is $180^\circ$, two right angles can form a linear pair. This statement is also TRUE.

Both statements (C) and (D) are true statements regarding the types of angles that can form a linear pair. However, typically in multiple-choice questions, there is only one best answer. The case described in (C) (an acute and an obtuse angle) occurs when two non-perpendicular lines intersect, forming linear pairs where the angles are unequal. The case described in (D) occurs specifically when two perpendicular lines intersect, forming linear pairs where the angles are equal. Statement (C) represents a more general scenario of supplementary angles formed by intersecting lines that are not necessarily perpendicular.

Assuming the question seeks a general characteristic of linear pairs formed by intersecting lines that are not necessarily perpendicular, option (C) is the intended answer.

The correct option is (C) An acute angle and an obtuse angle can form a linear pair.

Explanation:

Let's analyze the properties of each angle pair when two parallel lines are intersected by a transversal:

(i) Corresponding Angles: When a transversal intersects two parallel lines, corresponding angles are equal.

This matches property (c) "Are equal".

(ii) Alternate Interior Angles: When a transversal intersects two parallel lines, alternate interior angles are equal.

This matches property (c) "Are equal".

(iii) Interior angles on the same side: When a transversal intersects two parallel lines, the sum of the interior angles on the same side of the transversal is $180^\circ$. These angles are supplementary.

This matches property (a) "Sum is $180^\circ$".

(iv) Vertically Opposite Angles: When two lines intersect, the vertically opposite angles formed are always equal, regardless of whether the lines are parallel or not.

This matches property (b) "Always equal (regardless of parallel lines)".

Based on the analysis, the correct matches are:

(i) Corresponding Angles $\to$ (c) Are equal

(ii) Alternate Interior Angles $\to$ (c) Are equal

(iii) Interior angles on the same side $\to$ (a) Sum is $180^\circ$

(iv) Vertically Opposite Angles $\to$ (b) Always equal (regardless of parallel lines)

Now let's check the options:

(A) (i)-(c), (ii)-(c), (iii)-(a), (iv)-(b)

(B) (i)-(c), (ii)-(a), (iii)-(c), (iv)-(b)

(C) (i)-(a), (ii)-(c), (iii)-(c), (iv)-(b)

(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(c)

Comparing our matches with the options, option (A) correctly represents all the pairings.

The correct option is (A) (i)-(c), (ii)-(c), (iii)-(a), (iv)-(b).

Explanation:

Let's analyze the given Assertion (A) and Reason (R).

Assertion (A): If two angles form a linear pair, they are supplementary.

A linear pair consists of two adjacent angles whose non-common sides form a straight line. The sum of the angles that form a straight line is always $180^\circ$. Therefore, the sum of the measures of the two angles in a linear pair is $180^\circ$. Angles whose sum is $180^\circ$ are defined as supplementary angles.

Thus, Assertion (A) is True.

Reason (R): Supplementary angles are a pair of angles whose sum is $180^\circ$.

This statement is the standard definition of supplementary angles in geometry.

Thus, Reason (R) is True.

Now we consider whether Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states a property of linear pairs (being supplementary). Reason (R) provides the definition of supplementary angles. The definition of supplementary angles is precisely why angles in a linear pair are called supplementary, because their sum is $180^\circ$. The definition in R directly explains the characteristic stated in A.

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

Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Looking at the given options, this corresponds to option (A).

The correct option is (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Let's analyze the given Assertion (A) and Reason (R).

Assertion (A): If a transversal intersects two lines such that the sum of consecutive interior angles is $180^\circ$, then the two lines are parallel.

This statement is a well-known criterion for parallel lines. If the sum of the interior angles on the same side of the transversal is $180^\circ$, it implies that the lines are parallel. This is a fundamental theorem in geometry used to prove lines are parallel.

Thus, Assertion (A) is True.

Reason (R): This is the converse of a property of parallel lines.

Consider the original property: "If two lines are parallel, then the sum of the interior angles on the same side of the transversal is $180^\circ$."

The converse of a conditional statement "If P, then Q" is "If Q, then P".

In this case, let P be "two lines are parallel" and Q be "the sum of consecutive interior angles is $180^\circ$".

The converse statement is "If the sum of consecutive interior angles is $180^\circ$, then the two lines are parallel".

This is exactly what Assertion (A) states.

Therefore, Reason (R) is also a true statement, correctly identifying Assertion (A) as the converse of the property.

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

Assertion (A) is true because it is the converse of the property of parallel lines. The truth of Assertion (A) is directly established by the fact that it is the converse of a known geometric property that holds when lines are parallel. In this specific case, the converse of the property is also true and serves as a test for parallelism.

Hence, Reason (R) provides the correct theoretical basis for why Assertion (A) is true.

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).

Comparing this with the given options, option (A) is the correct choice.

The correct option is (A) Both A and R are true and R is the correct explanation of A.

Explanation:

This case study describes a geometric situation involving two lines (the long pieces of wood) intersected by a transversal line (the crossing piece of wood).

The problem states that the carpenter wants the two long pieces to be parallel. It gives the measure of one interior angle formed by the transversal and one of the long pieces on a certain side, and asks for the measure of the interior angle on the same side of the transversal with the other long piece, which is required for the two long pieces to be parallel.

The two angles mentioned are consecutive interior angles (or interior angles on the same side of the transversal).

Relevant Geometric Property:

A key property related to parallel lines and transversals states that if two lines are intersected by a transversal, the two lines are parallel if and only if the sum of the measures of the interior angles on the same side of the transversal is $180^\circ$. These angles are supplementary.

Calculation:

Let the measure of the given interior angle on one side of the transversal be $\alpha$. We are given $\alpha = 65^\circ$.

Let the measure of the interior angle on the same side of the transversal with the other line be $\beta$.

For the two long pieces of wood to be parallel, the sum of these consecutive interior angles must be $180^\circ$.

So, we have the equation:

$\alpha + \beta = 180^\circ$

Substitute the given value of $\alpha$:

$65^\circ + \beta = 180^\circ$

To find $\beta$, subtract $65^\circ$ from both sides:

$\beta = 180^\circ - 65^\circ$

$\beta = 115^\circ$

Therefore, to ensure the two long pieces are parallel, the crossing piece should make an angle of $115^\circ$ with the other long piece on the inside, on the same side of the crossing piece.

This matches option (B).

The correct option is (B) $115^\circ$.

Explanation:

A linear pair is a pair of adjacent angles formed when two lines intersect. The non-common sides of the adjacent angles in a linear pair form a straight line.

The property of a linear pair is that the sum of the measures of the two angles that form a linear pair is always equal to $180^\circ$. These angles are supplementary.

Let the measure of one angle in the linear pair be $\theta_1$, and the measure of the other angle be $\theta_2$.

We are given that one angle is $75^\circ$. Let's say $\theta_1 = 75^\circ$.

According to the property of a linear pair, the sum of the two angles is $180^\circ$:

$\theta_1 + \theta_2 = 180^\circ$

Substitute the given value of $\theta_1$ into the equation:

$75^\circ + \theta_2 = 180^\circ$

To find the measure of the other angle $\theta_2$, subtract $75^\circ$ from both sides of the equation:

$\theta_2 = 180^\circ - 75^\circ$

$\theta_2 = 105^\circ$

The measure of the other angle in the linear pair is $105^\circ$.

Comparing this result with the given options, we find that the measure of the other angle is $105^\circ$, which corresponds to option (B).

The correct option is (B) $105^\circ$.

Explanation:

In $\triangle PQR$, we are given:

The exterior angle at vertex R is $120^\circ$.

The measure of interior angle $\angle P$ is $70^\circ$.

We need to find the measure of interior angle $\angle Q$.

According to 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 opposite interior angles.

In $\triangle PQR$, the interior angles opposite to the exterior angle at R are $\angle P$ and $\angle Q$.

Applying the Exterior Angle Property:

Exterior angle at R = $\angle P + \angle Q$

Substitute the given values into this equation:

$120^\circ = 70^\circ + \angle Q$

To find the measure of $\angle Q$, subtract $70^\circ$ from both sides of the equation:

$\angle Q = 120^\circ - 70^\circ$

$\angle Q = 50^\circ$

The measure of angle $\angle Q$ is $50^\circ$. This matches option (A).

The correct option is (A) $50^\circ$.

Explanation:

We are considering two distinct lines that lie in the same plane.

Let's analyze the possible relationships between two lines in a plane:

1. Parallel lines: Two lines in a plane are parallel if they do not intersect at any point, no matter how far they are extended. The distance between parallel lines is constant.

2. Intersecting lines: Two lines in a plane are intersecting if they cross each other at exactly one point. Perpendicular lines are a specific type of intersecting lines where the angle of intersection is $90^\circ$.

Now, let's consider the given options:

(A) Perpendicular: Perpendicular lines are a specific case of intersecting lines. While possible for non-parallel lines, it's not the only alternative to being parallel.

(B) Coincident: Coincident lines are lines that occupy the exact same set of points. The question specifies "two distinct lines", meaning they are not the same line. So, they cannot be coincident.

(C) Intersecting: Intersecting lines cross at exactly one point. If two distinct lines in a plane are not parallel, they must intersect.

(D) Skew: Skew lines are lines that are neither parallel nor intersecting. This occurs when lines lie in different planes (they are non-coplanar). Since the question states the lines are "in a plane", they must be coplanar, and thus cannot be skew.

For two distinct lines lying in the same plane, there are only two possibilities: they either never meet (parallel) or they meet at exactly one point (intersecting).

Therefore, two distinct lines in a plane can either be parallel or intersecting.

The correct option is (C) Intersecting.

Explanation:

Two angles are said to be complementary if the sum of their measures is exactly $90^\circ$.

If an angle has a measure of $\theta$, its complement has a measure of $90^\circ - \theta$.

We are given an angle with a measure of $55^\circ$.

Let the given angle be $\theta_1 = 55^\circ$.

Let the complement of this angle be $\theta_2$.

According to the definition of complementary angles:

$\theta_1 + \theta_2 = 90^\circ$

Substitute the given value of $\theta_1$ into the equation:

$55^\circ + \theta_2 = 90^\circ$

To find the measure of the complement $\theta_2$, subtract $55^\circ$ from both sides of the equation:

$\theta_2 = 90^\circ - 55^\circ$

$\theta_2 = 35^\circ$

The complement of an angle of $55^\circ$ is $35^\circ$.

Comparing this result with the given options, we find that the measure of the complement is $35^\circ$, which corresponds to option (A).

The correct option is (A) $35^\circ$.

Explanation:

We are given that a transversal intersects two lines, and the sum of the interior angles on the same side of the transversal is not equal to $180^\circ$.

Let the two lines be $l$ and $m$, and let $t$ be the transversal. Let the interior angles on the same side of the transversal be $\angle 1$ and $\angle 2$. We are given that $\angle 1 + \angle 2 \neq 180^\circ$.

Let's recall the property of interior angles on the same side when two lines are parallel:

If two lines are parallel, then the sum of the interior angles on the same side of the transversal is $180^\circ$.

The converse of this property is also true:

If the sum of the interior angles on the same side of the transversal is $180^\circ$, then the two lines are parallel.

The question states that the sum of the interior angles on the same side is NOT $180^\circ$. This is the negation of the condition for parallel lines based on this property.

If the sum is not $180^\circ$, then the lines cannot be parallel.

In a plane, two distinct lines can have only two possible relationships: they are either parallel or they intersect at a single point.

Since the lines are not parallel, they must intersect.

Therefore, if a transversal intersects two lines such that the sum of the interior angles on the same side is not $180^\circ$, the lines are intersecting.

Comparing this conclusion with the given options, we find that option (C) is correct.

The correct option is (C) Intersecting.

Explanation:

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

To determine which sets of angles could be the angles of a triangle, we need to calculate the sum of the angles in each option and check if the sum is $180^\circ$.

Let's examine each option:

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

Sum of angles = $60^\circ + 60^\circ + 60^\circ = 180^\circ$.

Since the sum is $180^\circ$, these angles can form a triangle (specifically, an equilateral triangle).

(B) $90^\circ, 45^\circ, 45^\circ$

Sum of angles = $90^\circ + 45^\circ + 45^\circ = 180^\circ$.

Since the sum is $180^\circ$, these angles can form a triangle (specifically, a right-angled isosceles triangle).

(C) $30^\circ, 50^\circ, 100^\circ$

Sum of angles = $30^\circ + 50^\circ + 100^\circ = 180^\circ$.

Since the sum is $180^\circ$, these angles can form a triangle (specifically, an obtuse-angled triangle).

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

Sum of angles = $100^\circ + 40^\circ + 40^\circ = 180^\circ$.

Since the sum is $180^\circ$, these angles can form a triangle (specifically, an obtuse-angled isosceles triangle).

(E) $70^\circ, 80^\circ, 30^\circ$

Sum of angles = $70^\circ + 80^\circ + 30^\circ = 180^\circ$.

Since the sum is $180^\circ$, these angles can form a triangle (specifically, an acute-angled triangle).

In all the given options, the sum of the three angles is $180^\circ$. Therefore, all these sets of angles could form a triangle.

The correct options are (A), (B), (C), (D), and (E).

Given:

$\angle A$ and $\angle B$ form a linear pair.

$\angle A = 2\angle B$

To Find:

The measures of $\angle A$ and $\angle B$.

Solution:

Since $\angle A$ and $\angle B$ form a linear pair, the sum of their measures is $180^\circ$.

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

We are also given the relationship between $\angle A$ and $\angle B$:

$\angle A = 2\angle B$

Substitute the second equation into the first equation:

$(2\angle B) + \angle B = 180^\circ$

Combine the terms involving $\angle B$:

$3\angle B = 180^\circ$

Now, solve for $\angle B$ by dividing both sides by 3:

$\angle B = \frac{180^\circ}{3}$

$\angle B = 60^\circ$

Now that we have the measure of $\angle B$, we can use the given relationship $\angle A = 2\angle B$ to find the measure of $\angle A$:

$\angle A = 2 \times 60^\circ$

$\angle A = 120^\circ$

So, the measures of the angles are $\angle A = 120^\circ$ and $\angle B = 60^\circ$.

We can verify this by checking if they form a linear pair:

$\angle A + \angle B = 120^\circ + 60^\circ = 180^\circ$. This confirms they form a linear pair.

We can also check if $\angle A = 2\angle B$:

$120^\circ = 2 \times 60^\circ \implies 120^\circ = 120^\circ$. This confirms the given relationship.

Comparing our results with the given options:

(A) $\angle A = 60^\circ, \angle B = 120^\circ$ (Incorrect)

(B) $\angle A = 120^\circ, \angle B = 60^\circ$ (Correct)

(C) $\angle A = 90^\circ, \angle B = 45^\circ$ (Incorrect, sum is $135^\circ$)

(D) $\angle A = 30^\circ, \angle B = 150^\circ$ (Incorrect, relationship is $\angle B = 5\angle A$)

The correct option is (B) $\angle A = 120^\circ, \angle B = 60^\circ$.

Explanation:

Consider a line, say $AB$, and a ray, say $OC$, standing on the line at point $O$. Point $O$ is the vertex of the angles formed, and $OA$, $OB$ are parts of the line, while $OC$ is the common arm.

The two adjacent angles formed by the ray $OC$ standing on the line $AB$ are $\angle AOC$ and $\angle BOC$. These angles are adjacent because they share a common vertex $O$ and a common arm $OC$, and their non-common arms $OA$ and $OB$ lie on the same line $AB$.

Angles that are adjacent and whose non-common arms form a straight line are called a linear pair.

A fundamental postulate (often called the Linear Pair Postulate or Axiom) states that if a ray stands on a line, then the sum of the two adjacent angles so formed is $180^\circ$. This is because the angles form a straight angle, whose measure is $180^\circ$.

Thus, $\angle AOC + \angle BOC = 180^\circ$.

The sum of the two adjacent angles formed when a ray stands on a line is $180^\circ$.

Comparing this with the given options, the correct value is $180^\circ$.

The correct option is (B) $180^\circ$.

Explanation:

We are given that the angles of a triangle are in the ratio $1:2:3$.

Let the angles of the triangle be $x$, $2x$, and $3x$, where $x$ is a common multiple.

According to the Angle Sum Property of a Triangle, the sum of the interior angles of any triangle is $180^\circ$.

Calculation:

Sum of the angles = $x + 2x + 3x = 180^\circ$

Combine the terms on the left side:

$6x = 180^\circ$

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

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

$x = 30^\circ$

Now, substitute the value of $x$ back into the expressions for the angles:

First angle = $x = 30^\circ$

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

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

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

Comparing these angles with the given options:

(A) $30^\circ, 60^\circ, 90^\circ$ (Matches our calculated angles)

(B) $45^\circ, 45^\circ, 90^\circ$ (Sum = $180^\circ$, but ratio is $1:1:2$)

(C) $60^\circ, 60^\circ, 60^\circ$ (Sum = $180^\circ$, but ratio is $1:1:1$)

(D) $20^\circ, 40^\circ, 120^\circ$ (Sum = $180^\circ$, but ratio is $1:2:6$)

The set of angles that matches our calculated angles and the given ratio is $30^\circ, 60^\circ, 90^\circ$.

The correct option is (A) $30^\circ, 60^\circ, 90^\circ$.

Explanation:

Two angles are called supplementary angles if the sum of their measures is equal to $180^\circ$.

An angle bisector is a ray that divides an angle into two equal angles.

Let the two supplementary angles be $\angle AOC$ and $\angle BOC$. Since they are supplementary, their sum is $180^\circ$. We can assume they are adjacent angles that form a linear pair, as their bisectors meeting at a point and forming an angle between them implies a common vertex and likely adjacent positioning on a straight line.

Let $\angle AOC = \alpha$ and $\angle BOC = \beta$.

Then, $\alpha + \beta = 180^\circ$

Let $OD$ be the bisector of $\angle AOC$, and $OE$ be the bisector of $\angle BOC$. The point of intersection of the lines forming $\angle AOC$ and $\angle BOC$ is the common vertex, say $O$. $OA$ and $OB$ form a straight line, and $OC$ is the common arm.

Since $OD$ bisects $\angle AOC$, we have $\angle DOC = \frac{1}{2}\angle AOC = \frac{\alpha}{2}$.

Since $OE$ bisects $\angle BOC$, we have $\angle COE = \frac{1}{2}\angle BOC = \frac{\beta}{2}$.

The angle between the bisectors $OD$ and $OE$ is $\angle DOE$.

From the figure (mentally visualize or sketch), the angle $\angle DOE$ is the sum of the angles $\angle DOC$ and $\angle COE$ because $OC$ lies between $OD$ and $OE$ (assuming $OA$, $OC$, $OB$ are arranged in order on the straight line).

$\angle DOE = \angle DOC + \angle COE$

Substitute the values of $\angle DOC$ and $\angle COE$:

$\angle DOE = \frac{\alpha}{2} + \frac{\beta}{2}$

Combine the terms:

$\angle DOE = \frac{\alpha + \beta}{2}$

We know that $\alpha + \beta = 180^\circ$ (since the angles are supplementary):

$\angle DOE = \frac{180^\circ}{2}$

$\angle DOE = 90^\circ$

The angle between the bisectors of two supplementary angles is always $90^\circ$. This means the bisectors are perpendicular to each other.

Comparing this result with the given options, we find that the angle is $90^\circ$, which corresponds to option (B).

The correct option is (B) $90^\circ$.

Explanation:

When a transversal intersects two parallel lines, specific relationships hold true for certain pairs of angles.

Let's examine each type of angle pair mentioned in the options:

(A) Corresponding angles: When a transversal intersects two parallel lines, corresponding angles are equal.

(B) Alternate interior angles: When a transversal intersects two parallel lines, alternate interior angles are equal.

(C) Consecutive interior angles (or Interior angles on the same side): When a transversal intersects two parallel lines, the sum of the measures of consecutive interior angles is $180^\circ$. These angles are supplementary. They are not necessarily equal (unless both angles are $90^\circ$, which happens when the transversal is perpendicular to the parallel lines).

(D) Alternate exterior angles: When a transversal intersects two parallel lines, alternate exterior angles are equal.

The question asks which pair of angles is NOT necessarily equal when a transversal intersects two parallel lines.

Based on the properties, corresponding angles, alternate interior angles, and alternate exterior angles are necessarily equal.

Consecutive interior angles are supplementary (sum to $180^\circ$), but they are not necessarily equal.

Therefore, consecutive interior angles are the pair that is not necessarily equal when a transversal intersects two parallel lines.

The correct option is (C) Consecutive interior angles.

Given:

Lines $AB \parallel CD$.

A transversal intersects $AB$ and $CD$.

Two alternate interior angles formed are $75^\circ$ and $x$.

To Find:

The value of $x$.

Solution:

We are given that lines $AB$ and $CD$ are parallel ($AB \parallel CD$) and they are intersected by a transversal.

We are also given that the angle with measure $75^\circ$ and the angle with measure $x$ are alternate interior angles.

According to the property of parallel lines intersected by a transversal:

If two parallel lines are intersected by a transversal, then each pair of alternate interior angles is equal.

Since $AB \parallel CD$, the alternate interior angles must be equal.

$x = 75^\circ$

The value of $x$ is $75^\circ$.

Comparing this result with the given options, we find that the value of $x$ is $75^\circ$, which corresponds to option (A).

The correct option is (A) $75^\circ$.

Explanation:

Two angles are said to be complementary if the sum of their measures is $90^\circ$.

Let the measure of the angle be $x$.

The measure of its complement is $90^\circ - x$.

The problem states that the angle is equal to its complement.

So, we can write the equation:

$x = 90^\circ - x$

To solve for $x$, add $x$ to both sides of the equation:

$x + x = 90^\circ$

$2x = 90^\circ$

Now, divide both sides by 2:

$x = \frac{90^\circ}{2}$

$x = 45^\circ$

So, the measure of the angle is $45^\circ$. Its complement is $90^\circ - 45^\circ = 45^\circ$, which is equal to the angle itself.

The measure of an angle that is equal to its complement is $45^\circ$.

Comparing this result with the given options, we find that the measure is $45^\circ$, which corresponds to option (B).

The correct option is (B) $45^\circ$.

Explanation:

Two angles are said to be supplementary if the sum of their measures is $180^\circ$.

Let the measure of the angle be $y$.

The measure of its supplement is $180^\circ - y$.

The problem states that the angle is equal to its supplement.

So, we can write the equation:

$y = 180^\circ - y$

To solve for $y$, add $y$ to both sides of the equation:

$y + y = 180^\circ$

$2y = 180^\circ$

Now, divide both sides by 2:

$y = \frac{180^\circ}{2}$

$y = 90^\circ$

So, the measure of the angle is $90^\circ$. Its supplement is $180^\circ - 90^\circ = 90^\circ$, which is equal to the angle itself.

The measure of an angle that is equal to its supplement is $90^\circ$.

Comparing this result with the given options, we find that the measure is $90^\circ$, which corresponds to option (A).

The correct option is (A) $90^\circ$.

Given:

Lines $PQ$ and $RS$ intersect at point $O$.

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

To Find:

The measure of $\angle SOQ$.

Solution:

When two lines intersect, they form pairs of vertically opposite angles.

In the given figure, lines $PQ$ and $RS$ intersect at $O$. The pairs of vertically opposite angles formed are $\angle POR$ and $\angle SOQ$, and $\angle POS$ and $\angle ROQ$.

A fundamental geometric property states that vertically opposite angles are equal.

Therefore, $\angle POR$ is vertically opposite to $\angle SOQ$.

By the property of vertically opposite angles:

$\angle SOQ = \angle POR$

We are given that $\angle POR = 50^\circ$.

Substituting the given value:

$\angle SOQ = 50^\circ$

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

Comparing this result with the given options, we find that the measure is $50^\circ$, which corresponds to option (A).

The correct option is (A) $50^\circ$.

Explanation:

This case study describes a situation involving two parallel lines (Street A and Street B) intersected by a transversal line (Street C).

We are given that Street A is parallel to Street B ($A \parallel B$), and Street C is a transversal intersecting both streets.

We are given the measure of an angle between Street A and Street C as $110^\circ$. This angle is on one side of the transversal.

We are asked to find the measure of the corresponding angle between Street B and Street C.

Relevant Geometric Property:

One of the fundamental properties related to parallel lines intersected by a transversal states that:

If two parallel lines are intersected by a transversal, then each pair of corresponding angles is equal in measure.

Calculation:

Let the given angle between Street A and Street C be $\alpha$, so $\alpha = 110^\circ$.

Let the corresponding angle between Street B and Street C be $\beta$.

Since Street A is parallel to Street B and Street C is the transversal, the corresponding angles $\alpha$ and $\beta$ are equal.

$\beta = \alpha$

Substitute the given value of $\alpha$:

$\beta = 110^\circ$

Therefore, the measure of the corresponding angle between Street B and Street C should be $110^\circ$ to ensure that Street A and Street B are parallel.

Comparing this result with the given options, we find that the measure is $110^\circ$, which corresponds to option (B).

The correct option is (B) $110^\circ$.

Explanation:

A linear pair is a pair of adjacent angles formed when a ray stands on a line, or when two lines intersect. The non-common sides of the adjacent angles in a linear pair form a straight line.

The sum of the measures of angles on a straight line is always $180^\circ$. Since the non-common sides of a linear pair form a straight line, the sum of the measures of the two angles in a linear pair is also $180^\circ$.

An angle whose measure is exactly $180^\circ$ is defined as a straight angle.

Therefore, the sum of two angles forming a linear pair is always equal to the measure of a straight angle, which is $180^\circ$.

Comparing this with the given options:

(A) Right angle: $90^\circ$ (Incorrect)

(B) Acute angle: Less than $90^\circ$ (Incorrect)

(C) Obtuse angle: More than $90^\circ$ but less than $180^\circ$ (Incorrect)

(D) Straight angle: Exactly $180^\circ$ (Correct)

The correct option is (D) Straight angle.

Explanation:

We are given that the angles of a triangle are $2x$, $3x$, and $4x$.

According to the Angle Sum Property of a Triangle, the sum of the measures of the interior angles of any triangle is always $180^\circ$.

Calculation:

The sum of the angles of the triangle is equal to $180^\circ$.

$(2x) + (3x) + (4x) = 180^\circ$

Combine the terms involving $x$:

$(2 + 3 + 4)x = 180^\circ$

$9x = 180^\circ$

Now, solve for $x$ by dividing both sides of the equation by 9:

$x = \frac{180^\circ}{9}$

$x = 20^\circ$

The value of $x$ is $20^\circ$.

We can find the actual angles: $2(20^\circ) = 40^\circ$, $3(20^\circ) = 60^\circ$, $4(20^\circ) = 80^\circ$. The sum is $40^\circ + 60^\circ + 80^\circ = 180^\circ$, which is correct.

Comparing our result with the given options, we find that the value of $x$ is $20^\circ$, which corresponds to option (B).

The correct option is (B) $20^\circ$.

Explanation:

We are given two parallel lines and a third line (a transversal) that is perpendicular to one of these parallel lines.

Let the two parallel lines be $l_1$ and $l_2$, such that $l_1 \parallel l_2$. Let the transversal line be $t$.

We are given that the transversal $t$ is perpendicular to one of the parallel lines, say $l_1$. This means the angle formed by the intersection of $t$ and $l_1$ is $90^\circ$.

When a transversal intersects two parallel lines, certain angle relationships hold true.

Consider the corresponding angle between the transversal $t$ and the line $l_2$. Since $l_1 \parallel l_2$, the corresponding angles formed by the transversal are equal.

If the angle between $t$ and $l_1$ is $90^\circ$, the corresponding angle between $t$ and $l_2$ is also $90^\circ$.

Alternatively, consider the interior angles on the same side of the transversal. If one of the interior angles formed by $t$ and $l_1$ is $90^\circ$, its consecutive interior angle on the same side, formed by $t$ and $l_2$, must sum up to $180^\circ$ with it (since $l_1 \parallel l_2$). So, the other angle is $180^\circ - 90^\circ = 90^\circ$.

An angle of $90^\circ$ between two lines indicates that the lines are perpendicular.

Therefore, if a line is perpendicular to one of two parallel lines, it is also perpendicular to the other line.

The correct word to fill in the blank is "perpendicular".

The correct option is (B) Perpendicular.

Explanation:

The statement "Two lines are parallel if and only if..." requires a condition that is equivalent to the lines being parallel. This means the condition must be true when the lines are parallel, and conversely, if the condition is true, then the lines must be parallel.

Let's examine each of the given options in the context of a transversal intersecting two lines:

(A) Corresponding angles are supplementary.

When two lines are parallel, corresponding angles are equal, not supplementary (unless both angles are $90^\circ$). If corresponding angles were supplementary in general, the lines would not necessarily be parallel. Thus, this statement is incorrect.

(B) Alternate interior angles are complementary.

When two lines are parallel, alternate interior angles are equal, not complementary (unless both angles are $45^\circ$). If alternate interior angles were complementary in general, the lines would not necessarily be parallel. Thus, this statement is incorrect.

(C) Interior angles on the same side of the transversal are supplementary.

Property: If two lines are parallel, then the interior angles on the same side of the transversal are supplementary (sum is $180^\circ$). This statement is true when lines are parallel.

Converse: If the interior angles on the same side of the transversal are supplementary (sum is $180^\circ$), then the two lines are parallel. This converse statement is also true and is a standard test for parallelism.

Since both the statement and its converse are true, this condition is equivalent to the lines being parallel. Thus, this statement correctly completes the "if and only if" condition.

(D) Vertically opposite angles are supplementary.

Vertically opposite angles are always equal when two lines intersect, regardless of whether the lines being intersected by a transversal are parallel or not. They are supplementary only if they are both $90^\circ$. Their property does not determine the parallelism of the two lines being intersected by the transversal. Thus, this statement is incorrect.

The condition that correctly completes the statement "Two lines are parallel if and only if..." based on transversal properties is that the interior angles on the same side of the transversal are supplementary.

The correct option is (C) Interior angles on the same side of the transversal are supplementary.

Line Segment: A line segment is a part of a line that is bounded by two distinct endpoints. It has a definite length and does not extend infinitely in either direction.

For example, if A and B are two points on a line, the collection of all points on the line between A and B, including A and B, is called the line segment AB. It is denoted by $\overline{AB}$.

Ray: A ray is a part of a line that has one endpoint and extends infinitely in only one direction. It has no definite length.

For example, if A is a point and B is another point on a line, the ray AB starts at endpoint A and goes through B, extending infinitely beyond B. It is denoted by $\overrightarrow{AB}$. The endpoint is always mentioned first.

Line: A line is a straight one-dimensional figure that has no thickness and extends infinitely in both directions. It has no endpoints and no definite length.

A line can be named by a single lowercase letter (like line 'l') or by two distinct points on the line (like line AB, denoted by $\overleftrightarrow{AB}$).

Differences: The main differences between a line segment, a ray, and a line are based on their endpoints and length:

• A line segment has two endpoints and a definite length.
• A ray has one endpoint and extends infinitely in one direction, so it has no definite length.
• A line has no endpoints and extends infinitely in both directions, so it has no definite length.

Definition of an Angle:

An angle is formed when two rays originate from the same endpoint.

The amount of rotation from one ray to the other is the measure of the angle.

Vertex of an Angle:

The vertex of an angle is the common endpoint from which the two rays forming the angle originate.

It is the point where the two arms meet.

Arms (or Sides) of an Angle:

The arms (or sides) of an angle are the two rays that form the angle.

These rays start from the vertex and extend outwards.

Based on their measures, angles are classified as follows:

(a) $45^\circ$: This is an acute angle because its measure is greater than $0^\circ$ and less than $90^\circ$ ($0^\circ < 45^\circ < 90^\circ$).

(b) $120^\circ$: This is an obtuse angle because its measure is greater than $90^\circ$ and less than $180^\circ$ ($90^\circ < 120^\circ < 180^\circ$).

(c) $90^\circ$: This is a right angle because its measure is exactly $90^\circ$.

(d) $180^\circ$: This is a straight angle because its measure is exactly $180^\circ$.

Complementary Angles:

Two angles are said to be complementary if the sum of their measures is $90^\circ$.

If angle A and angle B are complementary, then $m(\angle A) + m(\angle B) = 90^\circ$.

Finding the other angle:

Given that two angles are complementary, and one angle measures $52^\circ$.

Let the unknown angle be $x$.

By the definition of complementary angles, the sum of the two angles must be $90^\circ$.

$x + 52^\circ = 90^\circ$

To find the value of $x$, we subtract $52^\circ$ from both sides of the equation:

$x = 90^\circ - 52^\circ$

$x = 38^\circ$

Thus, the measure of the other angle is $38^\circ$.

Supplementary Angles:

Two angles are said to be supplementary if the sum of their measures is $180^\circ$.

If angle C and angle D are supplementary, then $m(\angle C) + m(\angle D) = 180^\circ$.

Finding the supplement of $105^\circ$:

Given that one angle measures $105^\circ$.

We need to find its supplement.

Let the supplement of the $105^\circ$ angle be $y$.

By the definition of supplementary angles, the sum of the angle and its supplement must be $180^\circ$.

$105^\circ + y = 180^\circ$

To find the value of $y$, we subtract $105^\circ$ from both sides of the equation:

$y = 180^\circ - 105^\circ$

$y = 75^\circ$

Thus, the supplement of an angle measuring $105^\circ$ is $75^\circ$.

Adjacent Angles:

Two angles are called adjacent angles if they have a common vertex, a common arm, and their non-common arms are on opposite sides of the common arm.

For example, in the figure below (imagine a ray OC between rays OA and OB originating from O), $\angle \text{AOC}$ and $\angle \text{COB}$ are adjacent angles. They share vertex O, common arm OC, and non-common arms OA and OB are on opposite sides of OC.

Linear Pair:

A linear pair is a pair of adjacent angles whose non-common arms are opposite rays.

When the non-common arms form a straight line, the two adjacent angles form a linear pair.

The sum of the measures of angles in a linear pair is always $180^\circ$.

For example, if ray OC stands on line AB (imagine O is on the line AB, and C is a point not on the line), then $\angle \text{AOC}$ and $\angle \text{BOC}$ form a linear pair. OA and OB are opposite rays forming the straight line AB. $\angle \text{AOC} + \angle \text{BOC} = 180^\circ$.

Linear pair as a special case of adjacent angles:

A linear pair is a special type of adjacent angle pair because it satisfies all the conditions of adjacent angles:

• They have a common vertex.
• They have a common arm.
• Their non-common arms are on opposite sides of the common arm.

In addition to being adjacent, a linear pair has the specific characteristic that their non-common arms form a straight line, meaning they are opposite rays. This results in the sum of the angles being $180^\circ$.

Therefore, every linear pair is a pair of adjacent angles, but not every pair of adjacent angles is a linear pair.

Vertically Opposite Angles:

When two lines intersect each other, they form four angles at the point of intersection. The pairs of angles that are opposite to each other at the intersection point are called vertically opposite angles.

Imagine two lines, say line AB and line CD, intersecting at a point O. This forms four angles: $\angle \text{AOC}$, $\angle \text{AOD}$, $\angle \text{BOC}$, and $\angle \text{BOD}$. The pairs of vertically opposite angles are $(\angle \text{AOC}, \angle \text{BOD})$ and $(\angle \text{AOD}, \angle \text{BOC})$.

Relationship between Vertically Opposite Angles:

The relationship between vertically opposite angles is that they are always equal in measure.

So, if lines AB and CD intersect at O, then $m(\angle \text{AOC}) = m(\angle \text{BOD})$ and $m(\angle \text{AOD}) = m(\angle \text{BOC})$.

Finding the measure of the vertically opposite angle:

Given that one angle formed by the intersection of two lines is $70^\circ$.

Let this angle be $\angle 1 = 70^\circ$. Its vertically opposite angle is $\angle 3$.

According to the property of vertically opposite angles, they are equal.

$m(\angle 3) = m(\angle 1)$

$m(\angle 3) = 70^\circ$

Therefore, the measure of the vertically opposite angle to the $70^\circ$ angle is $70^\circ$.

Given:

$\angle 1 = 50^\circ$

$\angle 1$ and $\angle 2$ form a linear pair.

To Find:

The measure of $\angle 2$.

Solution:

We are given that $\angle 1$ and $\angle 2$ form a linear pair.

By the property of linear pairs, the sum of the measures of the two angles is $180^\circ$.

$m(\angle 1) + m(\angle 2) = 180^\circ$

Substitute the given value of $m(\angle 1) = 50^\circ$ into the equation:

$50^\circ + m(\angle 2) = 180^\circ$

To find $m(\angle 2)$, subtract $50^\circ$ from both sides of the equation:

$m(\angle 2) = 180^\circ - 50^\circ$

$m(\angle 2) = 130^\circ$

Thus, the measure of $\angle 2$ is $130^\circ$.

Given:

$\angle 1 = 50^\circ$

$\angle 1$ and $\angle 3$ are vertically opposite angles.

To Find:

The measure of $\angle 3$.

Solution:

We are given that $\angle 1$ and $\angle 3$ are vertically opposite angles.

By the property of vertically opposite angles, vertically opposite angles are equal in measure.

$m(\angle 3) = m(\angle 1)$

Substitute the given value of $m(\angle 1) = 50^\circ$ into the equation:

$m(\angle 3) = 50^\circ$

Thus, the measure of $\angle 3$ is $50^\circ$.

Transversal Line:

A transversal is a line that intersects two or more distinct lines at different points.

When a transversal intersects two lines, it creates several angles, including interior angles, exterior angles, corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles (or co-interior angles), and consecutive exterior angles.

Diagram Showing a Transversal Intersecting Two Lines:

To show this in a diagram, draw two lines, say line 'l' and line 'm'. These lines can be parallel or non-parallel.

Then, draw a third line, say line 't', that crosses both line 'l' and line 'm'.

Ensure that line 't' intersects line 'l' at one point (let's call it P) and intersects line 'm' at a different point (let's call it Q).

Line 't' is the transversal.

The diagram would look something like this (imagine lines drawn as described):

Line l

\

\ P

\

------t-------

\

\ Q

\

Line m

Here, line 't' intersects line 'l' at point P and line 'm' at point Q. Therefore, 't' is a transversal to lines 'l' and 'm'.

Let the two lines be 'l' and 'm', and the transversal be 't', intersecting line 'l' at point P and line 'm' at point Q. Let the angles formed be numbered 1, 2, 3, 4 around point P (say, starting from top-left and going clockwise) and angles 5, 6, 7, 8 around point Q (similarly, starting from top-left and going clockwise).

Angle 1 is at the top left of the intersection at P. Angle 5 is at the top left of the intersection at Q.

Angle 2 is at the top right at P. Angle 6 is at the top right at Q.

Angle 3 is at the bottom right at P. Angle 7 is at the bottom right at Q.

Angle 4 is at the bottom left at P. Angle 8 is at the bottom left at Q.

Corresponding Angles:

Corresponding angles are pairs of angles that are in the same relative position at each intersection where a straight line crosses two others.

They occupy the same corner at each intersection.

A pair of corresponding angles is $\angle 1$ and $\angle 5$.

Other pairs of corresponding angles in this figure are $(\angle 2, \angle 6)$, $(\angle 3, \angle 7)$, and $(\angle 4, \angle 8)$.

Alternate Interior Angles:

Alternate interior angles are pairs of angles on opposite sides of the transversal and between the two lines that are intersected.

A pair of alternate interior angles is $\angle 4$ and $\angle 6$.

The other pair of alternate interior angles in this figure is $(\angle 3, \angle 5)$.

Using the same angle numbering convention as described in the answer to Question 11 (angles 1-4 at the first intersection, angles 5-8 at the second, clockwise from top-left):

Alternate Exterior Angles:

Alternate exterior angles are pairs of angles that are on opposite sides of the transversal and outside the two lines.

A pair of alternate exterior angles is $\angle 1$ and $\angle 7$.

The other pair of alternate exterior angles is $(\angle 2, \angle 8)$.

Consecutive Interior Angles (Interior angles on the same side of the transversal):

Consecutive interior angles are pairs of angles that are on the same side of the transversal and between the two lines.

A pair of consecutive interior angles is $\angle 4$ and $\angle 5$.

The other pair of consecutive interior angles is $(\angle 3, \angle 6)$.

Relationship between corresponding angles when a transversal intersects two parallel lines:

When a transversal intersects two parallel lines, each pair of corresponding angles is equal in measure.

For example, if line 'l' is parallel to line 'm', and line 't' is a transversal intersecting them, then any pair of corresponding angles (like $\angle 1$ and $\angle 5$ in the diagram used in Question 11) will have the same measure.

Finding the measure of the other corresponding angle:

Given that a transversal intersects two parallel lines, and one corresponding angle measures $80^\circ$.

Let the two corresponding angles be $\angle A$ and $\angle B$. We are given $m(\angle A) = 80^\circ$.

Since the lines are parallel, the corresponding angles are equal in measure.

$m(\angle B) = m(\angle A)$

Substitute the given measure of $\angle A$:

$m(\angle B) = 80^\circ$

Therefore, the measure of the other corresponding angle is $80^\circ$.

Relationship between alternate interior angles when a transversal intersects two parallel lines:

When a transversal intersects two parallel lines, each pair of alternate interior angles is equal in measure.

For example, in the diagram used in Question 11, if line 'l' is parallel to line 'm', then $m(\angle 4) = m(\angle 6)$ and $m(\angle 3) = m(\angle 5)$.

Given:

A transversal intersects two parallel lines.

One alternate interior angle measures $110^\circ$.

To Find:

The measure of the other alternate interior angle.

Solution:

Let the two alternate interior angles be $\angle X$ and $\angle Y$. We are given $m(\angle X) = 110^\circ$.

Since the lines are parallel, the alternate interior angles are equal in measure.

$m(\angle Y) = m(\angle X)$

Substitute the given measure of $\angle X$:

$m(\angle Y) = 110^\circ$

Therefore, the measure of the other alternate interior angle is $110^\circ$.

Given:

A transversal intersects two lines such that the sum of a pair of consecutive interior angles is $180^\circ$.

Let the two lines be 'l' and 'm', and the transversal be 't'. Let $\angle 4$ and $\angle 5$ be a pair of consecutive interior angles (using the numbering from Question 11 figure).

Given $m(\angle 4) + m(\angle 5) = 180^\circ$.

Conclusion about the two lines:

If a transversal intersects two lines such that the sum of consecutive interior angles on the same side of the transversal is $180^\circ$, then the two lines are parallel.

Theorem supporting this conclusion:

This conclusion is supported by the Converse of the Consecutive Interior Angles Theorem (also known as the Converse of the Co-interior Angles Theorem).

Statement of the Theorem:

The Converse of the Consecutive Interior Angles Theorem states that if a transversal intersects two lines such that the interior angles on the same side of the transversal are supplementary (i.e., their sum is $180^\circ$), then the two lines are parallel.

Given:

Line $l$ is parallel to line $m$ ($l \parallel m$).

Line $t$ is a transversal intersecting lines $l$ and $m$.

$m(\angle 1) = 70^\circ$.

To Find:

The measure of $\angle 8$ ($m(\angle 8)$).

Solution:

In the given figure, $\angle 1$ and $\angle 8$ are a pair of alternate exterior angles formed by the transversal $t$ intersecting lines $l$ and $m$.

We know that if a transversal intersects two parallel lines, then each pair of alternate exterior angles is equal in measure.

Since $l \parallel m$, we have:

$m(\angle 8) = m(\angle 1)$

Substitute the given value $m(\angle 1) = 70^\circ$:

$m(\angle 8) = 70^\circ$

Thus, the measure of $\angle 8$ is $70^\circ$.

Alternate Solution:

$\angle 1$ and $\angle 5$ are corresponding angles. Since $l \parallel m$, corresponding angles are equal.

$m(\angle 5) = m(\angle 1) = 70^\circ$

$\angle 5$ and $\angle 8$ are vertically opposite angles. Vertically opposite angles are equal.

$m(\angle 8) = m(\angle 5)$

Substitute the value of $m(\angle 5)$:

$m(\angle 8) = 70^\circ$

Both methods yield the same result.

Given:

Line $l$ is parallel to line $m$ ($l \parallel m$).

Line $t$ is a transversal intersecting lines $l$ and $m$.

$m(\angle 1) = 70^\circ$.

To Find:

The measure of $\angle 4$ ($m(\angle 4)$).

Solution:

In the given figure, angles $\angle 1$ and $\angle 4$ are adjacent angles that form a straight line on line $l$ when intersected by the transversal $t$. This means $\angle 1$ and $\angle 4$ form a linear pair.

By the Linear Pair Axiom, the sum of the measures of angles in a linear pair is always $180^\circ$.

So, we have:

$m(\angle 1) + m(\angle 4) = 180^\circ$

Substitute the given value of $m(\angle 1) = 70^\circ$ into the equation:

$70^\circ + m(\angle 4) = 180^\circ$

To find $m(\angle 4)$, subtract $70^\circ$ from both sides of the equation:

$m(\angle 4) = 180^\circ - 70^\circ$

$m(\angle 4) = 110^\circ$

Thus, the measure of $\angle 4$ is $110^\circ$.

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 $180^\circ$.

If A, B, and C are the three interior angles of a triangle, then $m(\angle A) + m(\angle B) + m(\angle C) = 180^\circ$.

Given:

The angles of a triangle are in the ratio $2:3:4$.

To Find:

The measure of each angle of the triangle.

Solution:

Let the angles of the triangle be $2x$, $3x$, and $4x$, where $x$ is a common multiple.

By the Angle Sum Property of a triangle, the sum of the interior angles of a triangle is $180^\circ$.

Therefore, we can write the equation:

$2x + 3x + 4x = 180^\circ$

Combine the terms on the left side:

$(2+3+4)x = 180^\circ$

$9x = 180^\circ$

Divide both sides by 9 to solve for $x$:

$x = \frac{180^\circ}{9}$

$x = 20^\circ$

Now, substitute the value of $x$ back into the expressions for the angles:

First angle $= 2x = 2 \times 20^\circ = 40^\circ$

Second angle $= 3x = 3 \times 20^\circ = 60^\circ$

Third angle $= 4x = 4 \times 20^\circ = 80^\circ$

To verify, check if the sum of the angles is $180^\circ$:

$40^\circ + 60^\circ + 80^\circ = 180^\circ$

The sum is $180^\circ$, which confirms the values are correct.

Thus, the measures of the angles of the triangle are $40^\circ$, $60^\circ$, and $80^\circ$.

Exterior Angle Property of a Triangle:

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

Let's consider a triangle ABC. If one side, say BC, is extended to a point D, then $\angle \text{ACD}$ is an exterior angle.

The two interior angles opposite to the exterior angle $\angle \text{ACD}$ are $\angle \text{BAC}$ (or $\angle A$) and $\angle \text{ABC}$ (or $\angle B$).

According to the Exterior Angle Property:

$m(\angle \text{ACD}) = m(\angle \text{BAC}) + m(\angle \text{ABC})$

or simply,

$m(\angle \text{exterior}) = m(\angle \text{interior opposite angle 1}) + m(\angle \text{interior opposite angle 2})$

Given:

Measure of an exterior angle of a triangle $= 130^\circ$.

Measure of one interior opposite angle $= 60^\circ$.

To Find:

The measure of the other interior opposite angle.

The measure of the interior adjacent angle.

Solution:

Let the triangle be ABC, and let the exterior angle at vertex C be $130^\circ$. Let the interior opposite angles be $\angle A$ and $\angle B$. We are given one of these is $60^\circ$, say $\angle A = 60^\circ$. Let the other interior opposite angle be $\angle B$. Let the interior angle adjacent to the exterior angle at C be $\angle \text{ACB}$.

By the Exterior Angle Property of a triangle, the measure of an exterior angle is equal to the sum of the measures of its two interior opposite angles.

Exterior angle at C = $\angle A + \angle B$

$130^\circ = 60^\circ + \angle B$

Subtract $60^\circ$ from both sides to find $\angle B$ (the other interior opposite angle):

$\angle B = 130^\circ - 60^\circ$

$\angle B = 70^\circ$

The exterior angle at C and the interior adjacent angle $\angle \text{ACB}$ form a linear pair.

By the Linear Pair Axiom, the sum of their measures is $180^\circ$.

Exterior angle at C + $\angle \text{ACB} = 180^\circ$

$130^\circ + \angle \text{ACB} = 180^\circ$

Subtract $130^\circ$ from both sides to find $\angle \text{ACB}$ (the interior adjacent angle):

$\angle \text{ACB} = 180^\circ - 130^\circ$

$\angle \text{ACB} = 50^\circ$

Thus, the measure of the other interior opposite angle is $70^\circ$, and the measure of the interior adjacent angle is $50^\circ$.

Alternate Solution:

First, find the interior adjacent angle using the Linear Pair Axiom:

Interior adjacent angle $= 180^\circ - \text{Exterior angle}$

Interior adjacent angle $= 180^\circ - 130^\circ = 50^\circ$

Now, we have the three interior angles of the triangle: one is $60^\circ$, one is the interior adjacent angle we just found ($50^\circ$), and the third is the unknown interior opposite angle (let's call it $\angle x$).

By the Angle Sum Property of a triangle, the sum of the interior angles is $180^\circ$.

$60^\circ + 50^\circ + \angle x = 180^\circ$

Simplify and solve for $\angle x$:

$110^\circ + \angle x = 180^\circ$

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

$\angle x = 70^\circ$

Thus, the other interior opposite angle is $70^\circ$, and the interior adjacent angle is $50^\circ$. Both methods yield the same result.

Given:

Two distinct lines, say line $l$ and line $m$.

A third line, say line $t$, such that line $l$ is perpendicular to line $t$, and line $m$ is perpendicular to line $t$.

To Conclude:

Whether lines $l$ and $m$ are parallel to each other.

Justification (Solution):

Let line $l$ and line $m$ be intersected by the transversal line $t$.

We are given that $l \perp t$ and $m \perp t$.

When a line is perpendicular to another line, the angle formed at their intersection is a right angle, which measures $90^\circ$.

Let's consider the angles formed by the transversal $t$ with lines $l$ and $m$. Specifically, let's look at the interior angles on the same side of the transversal.

Let $\angle 1$ be one of the interior angles formed by line $l$ and transversal $t$, and let $\angle 2$ be the corresponding interior angle on the same side of the transversal formed by line $m$ and transversal $t$.

Since $l \perp t$, the angles formed at their intersection are $90^\circ$. Thus, $m(\angle 1) = 90^\circ$.

Since $m \perp t$, the angles formed at their intersection are $90^\circ$. Thus, $m(\angle 2) = 90^\circ$.

Now, consider the sum of these consecutive interior angles ($\angle 1$ and $\angle 2$).

$m(\angle 1) + m(\angle 2) = 90^\circ + 90^\circ = 180^\circ$

We have a pair of consecutive interior angles whose sum is $180^\circ$.

According to the Converse of the Consecutive Interior Angles Theorem, if a transversal intersects two lines such that the interior angles on the same side of the transversal are supplementary, then the two lines are parallel.

Since the sum of the consecutive interior angles ($m(\angle 1) + m(\angle 2)$) is $180^\circ$, we can conclude that line $l$ is parallel to line $m$.

Conclusion:

Yes, if two lines are perpendicular to the same line, they are parallel to each other.

This is justified by the Converse of the Consecutive Interior Angles Theorem.

Given:

Measure of the exterior angle of the triangle $= 120^\circ$.

Measure of one interior opposite angle $= 50^\circ$.

Measure of the other interior opposite angle $= x$.

To Find:

The value of $x$.

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 its two interior opposite angles.

According to this property, we can write the equation:

$120^\circ = 50^\circ + x$

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

$x = 120^\circ - 50^\circ$

$x = 70^\circ$

Thus, the value of $x$ is $70^\circ$.

Alternate Solution:

Let the interior adjacent angle to the exterior angle ($120^\circ$) be $y$.

The exterior angle and the interior adjacent angle form a linear pair, so their sum is $180^\circ$.

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

Solving for $y$:

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

$y = 60^\circ$

Now, consider the sum of the interior angles of the triangle. The interior angles are $50^\circ$, $x$, and $y$.

By the Angle Sum Property of a triangle, the sum of the interior angles is $180^\circ$.

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

Substitute the value of $y = 60^\circ$ into the equation:

$50^\circ + x + 60^\circ = 180^\circ$

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

Solving for $x$:

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

$x = 70^\circ$

Both methods give the same result for $x$.

Given:

Two lines intersect.

One angle formed is $40^\circ$.

An adjacent angle is labelled $y$.

The angle vertically opposite to $40^\circ$ is labelled $x$.

To Find:

The values of $x$ and $y$.

Solution:

First, let's find the value of $x$.

The angles labelled $40^\circ$ and $x$ are vertically opposite angles formed by the intersection of the two lines.

We know that vertically opposite angles are equal.

Therefore, $x = 40^\circ$.

Next, let's find the value of $y$.

The angles labelled $40^\circ$ and $y$ are adjacent angles that form a straight line.

These two angles form a linear pair.

We know that the sum of angles in a linear pair is $180^\circ$.

So, we have the equation: $40^\circ + y = 180^\circ$.

Subtract $40^\circ$ from both sides of the equation to find $y$:

$y = 180^\circ - 40^\circ$

$y = 140^\circ$

Thus, the value of $x$ is $40^\circ$ and the value of $y$ is $140^\circ$.

Given:

Let the first line be $l_1$, the second line be $l_2$, and the third line be $l_3$.

We are given that $l_1 \parallel l_2$ (the first line is parallel to the second line).

We are also given that $l_2 \perp l_3$ (the second line is perpendicular to the third line).

To Find:

The relationship between the first line ($l_1$) and the third line ($l_3$).

Solution:

Consider the third line $l_3$ as a transversal intersecting the second line $l_2$.

Since $l_2 \perp l_3$, the angle formed between lines $l_2$ and $l_3$ is $90^\circ$.

Now, consider the same line $l_3$ as a transversal intersecting the first line $l_1$ and the second line $l_2$. We know that $l_1 \parallel l_2$.

When a transversal intersects two parallel lines, the relationship between the angles formed can be used.

One such property states that if a transversal is perpendicular to one of two parallel lines, then it is also perpendicular to the other line.

Alternatively, we can consider corresponding angles. Let the intersection point of $l_2$ and $l_3$ form an angle of $90^\circ$. Since $l_1 \parallel l_2$, the corresponding angle formed by the intersection of $l_1$ and $l_3$ must be equal to the angle formed by $l_2$ and $l_3$.

Thus, the corresponding angle between $l_1$ and $l_3$ is also $90^\circ$.

An angle of $90^\circ$ signifies that the lines are perpendicular.

Conclusion:

The relationship between the first line ($l_1$) and the third line ($l_3$) is that the first line is perpendicular to the third line ($l_1 \perp l_3$).

Given:

Two lines AB and CD intersect each other at the point O.

To Prove:

The vertically opposite angles are equal, i.e., $m(\angle \text{AOC}) = m(\angle \text{BOD})$ and $m(\angle \text{AOD}) = m(\angle \text{BOC})$.

Diagram:

Imagine two straight lines, AB and CD, crossing each other at a point O. This forms four angles around O: $\angle \text{AOC}$ (formed by rays OA and OC), $\angle \text{AOD}$ (formed by rays OA and OD), $\angle \text{BOD}$ (formed by rays OB and OD), and $\angle \text{BOC}$ (formed by rays OB and OC). The pairs of vertically opposite angles are $\angle \text{AOC}$ and $\angle \text{BOD}$, and $\angle \text{AOD}$ and $\angle \text{BOC}$.

Proof:

Consider ray OA standing on the line CD.

The angles $\angle \text{AOC}$ and $\angle \text{AOD}$ form a linear pair.

Now, consider ray OD standing on the line AB.

The angles $\angle \text{AOD}$ and $\angle \text{BOD}$ form a linear pair.

From the two equations above, since both sums are equal to $180^\circ$, we can equate them:

$m(\angle \text{AOC}) + m(\angle \text{AOD}) = m(\angle \text{AOD}) + m(\angle \text{BOD})$

Subtract $m(\angle \text{AOD})$ from both sides of the equation:

$m(\angle \text{AOC}) = m(\angle \text{BOD})$

This proves that one pair of vertically opposite angles is equal.

Similarly, let's consider ray OC standing on the line AB.

The angles $\angle \text{AOC}$ and $\angle \text{BOC}$ form a linear pair.

From $m(\angle \text{AOC}) + m(\angle \text{AOD}) = 180^\circ$ and $m(\angle \text{AOC}) + m(\angle \text{BOC}) = 180^\circ$, we can equate them:

$m(\angle \text{AOC}) + m(\angle \text{AOD}) = m(\angle \text{AOC}) + m(\angle \text{BOC})$

Subtract $m(\angle \text{AOC})$ from both sides of the equation:

$m(\angle \text{AOD}) = m(\angle \text{BOC})$

This proves that the other pair of vertically opposite angles is equal.

Therefore, if two lines intersect each other, the vertically opposite angles are equal.

Hence, Proved.

Given:

A triangle ABC.

To Prove:

The sum of the interior angles of triangle ABC is $180^\circ$. That is, $m(\angle \text{ABC}) + m(\angle \text{BCA}) + m(\angle \text{BAC}) = 180^\circ$.

Diagram:

Imagine a triangle with vertices labelled A, B, and C. Draw a straight line passing through the vertex A such that this line is parallel to the side BC. Let this line be labelled DE, where D is a point on the line to the left of A, and E is a point on the line to the right of A. The line segment AB is a transversal intersecting DE and BC, and the line segment AC is another transversal intersecting DE and BC.

Construction:

Draw a line DE passing through vertex A parallel to the side BC.

Proof:

Since DE is a straight line, the angles formed on this line at point A sum up to $180^\circ$. These angles are $\angle \text{DAB}$, $\angle \text{BAC}$ (which is $\angle A$), and $\angle \text{EAC}$.

$m(\angle \text{DAB}) + m(\angle \text{BAC}) + m(\angle \text{EAC}) = 180^\circ$

Now, consider the parallel lines DE and BC, and the transversal AB.

The angles $\angle \text{DAB}$ and $\angle \text{ABC}$ (which is $\angle B$) are alternate interior angles.

Since DE $\parallel$ BC, alternate interior angles are equal.

$m(\angle \text{DAB}) = m(\angle \text{ABC})$

Next, consider the parallel lines DE and BC, and the transversal AC.

The angles $\angle \text{EAC}$ and $\angle \text{BCA}$ (which is $\angle C$) are alternate interior angles.

Since DE $\parallel$ BC, alternate interior angles are equal.

$m(\angle \text{EAC}) = m(\angle \text{BCA})$

Substitute the measures of $\angle \text{DAB}$ and $\angle \text{EAC}$ from the alternate interior angle properties into the equation for the straight line angles:

$m(\angle \text{ABC}) + m(\angle \text{BAC}) + m(\angle \text{BCA}) = 180^\circ$

This equation shows that the sum of the interior angles of triangle ABC ($\angle \text{A}$, $\angle \text{B}$, and $\angle \text{C}$) is equal to $180^\circ$.

Thus, the sum of the angles of a triangle is $180^\circ$.

Hence, Proved.

Given:

Two parallel lines $l$ and $m$ are intersected by a transversal line $t$.

To Prove:

Each pair of alternate interior angles is equal.

Referring to a standard diagram where line $t$ intersects line $l$ at P and line $m$ at Q, and interior angles are labelled 3, 4, 5, 6 (where 3, 4 are on one side of the transversal between the parallel lines, and 5, 6 are on the other side), we need to prove $m(\angle 4) = m(\angle 6)$ and $m(\angle 3) = m(\angle 5)$.

Diagram:

Imagine two horizontal parallel lines labelled $l$ and $m$. Draw a line labelled $t$ intersecting both lines at distinct points, say P on line $l$ and Q on line $m$. Label the angles formed. Let the upper intersection (at P) have angles 1, 2, 3, 4 (say, 1 top-left, 2 top-right, 3 bottom-right, 4 bottom-left). Let the lower intersection (at Q) have angles 5, 6, 7, 8 (say, 5 top-left, 6 top-right, 7 bottom-right, 8 bottom-left). The interior angles are $\angle 3, \angle 4$ (at P) and $\angle 5, \angle 6$ (at Q). The alternate interior angle pairs are $(\angle 4, \angle 6)$ and $(\angle 3, \angle 5)$.

Proof for $m(\angle 4) = m(\angle 6)$:

Angles $\angle 2$ and $\angle 4$ are vertically opposite angles at point P.

Lines $l$ and $m$ are parallel, and $t$ is a transversal. Angles $\angle 2$ and $\angle 6$ are corresponding angles.

From the above two equalities, since both $\angle 4$ and $\angle 6$ are equal to $\angle 2$, they must be equal to each other.

$m(\angle 4) = m(\angle 6)$

Proof for $m(\angle 3) = m(\angle 5)$:

Angles $\angle 1$ and $\angle 3$ are vertically opposite angles at point P.

Lines $l$ and $m$ are parallel, and $t$ is a transversal. Angles $\angle 1$ and $\angle 5$ are corresponding angles.

From the above two equalities, since both $\angle 3$ and $\angle 5$ are equal to $\angle 1$, they must be equal to each other.

$m(\angle 3) = m(\angle 5)$

Therefore, if a transversal intersects two parallel lines, each pair of alternate interior angles is equal.

Hence, Proved.

Given:

Two parallel lines $l$ and $m$ are intersected by a transversal line $t$.

To Prove:

Each pair of consecutive interior angles is supplementary (i.e., their sum is $180^\circ$).

Referring to a standard diagram where line $t$ intersects line $l$ at P and line $m$ at Q, and interior angles are labelled 3, 4, 5, 6 (where 3, 4 are on one side of the transversal between the parallel lines, and 5, 6 are on the other side), we need to prove $m(\angle 4) + m(\angle 5) = 180^\circ$ and $m(\angle 3) + m(\angle 6) = 180^\circ$.

Diagram:

Imagine two horizontal parallel lines labelled $l$ and $m$. Draw a line labelled $t$ intersecting both lines at distinct points, say P on line $l$ and Q on line $m$. Label the angles formed. Let the upper intersection (at P) have angles 1, 2, 3, 4 (say, 1 top-left, 2 top-right, 3 bottom-right, 4 bottom-left). Let the lower intersection (at Q) have angles 5, 6, 7, 8 (say, 5 top-left, 6 top-right, 7 bottom-right, 8 bottom-left). The consecutive interior angle pairs are $(\angle 4, \angle 5)$ and $(\angle 3, \angle 6)$.

Proof for $m(\angle 4) + m(\angle 5) = 180^\circ$:

Angles $\angle 2$ and $\angle 4$ form a linear pair on line $l$.

Lines $l$ and $m$ are parallel, and $t$ is a transversal. Angles $\angle 2$ and $\angle 5$ are corresponding angles.

Substitute $m(\angle 5)$ for $m(\angle 2)$ in the linear pair equation:

$m(\angle 5) + m(\angle 4) = 180^\circ$

or

$m(\angle 4) + m(\angle 5) = 180^\circ$

This shows that one pair of consecutive interior angles is supplementary.

Proof for $m(\angle 3) + m(\angle 6) = 180^\circ$:

Angles $\angle 1$ and $\angle 3$ form a linear pair on line $l$.

Lines $l$ and $m$ are parallel, and $t$ is a transversal. Angles $\angle 1$ and $\angle 6$ are corresponding angles.

Substitute $m(\angle 6)$ for $m(\angle 1)$ in the linear pair equation:

$m(\angle 6) + m(\angle 3) = 180^\circ$

or

$m(\angle 3) + m(\angle 6) = 180^\circ$

This shows that the other pair of consecutive interior angles is supplementary.

Therefore, if a transversal intersects two parallel lines, each pair of consecutive interior angles is supplementary.

Hence, Proved.

Alternative Proof using Alternate Interior Angles:

Consider the pair of consecutive interior angles $\angle 4$ and $\angle 5$.

Angles $\angle 4$ and $\angle 6$ are alternate interior angles. Since $l \parallel m$, we have $m(\angle 4) = m(\angle 6)$.

Angles $\angle 5$ and $\angle 6$ form a linear pair on line $m$.

Substitute $m(\angle 4)$ for $m(\angle 6)$ in the above equation (since $m(\angle 4) = m(\angle 6)$):

$m(\angle 5) + m(\angle 4) = 180^\circ$

or

$m(\angle 4) + m(\angle 5) = 180^\circ$

Similarly, $m(\angle 3) + m(\angle 5) = 180^\circ$ (linear pair) and $m(\angle 3) = m(\angle 5)$ (alternate interior angles) proves $m(\angle 3) + m(\angle 6) = 180^\circ$ by substitution.

This also proves the theorem.

Given:

Line $AB$ is parallel to line $CD$ ($AB \parallel CD$).

Line segment $EF$ is perpendicular to line $CD$ at point $E$ ($EF \perp CD$).

The measure of angle $\angle GED$ is $126^\circ$ ($m(\angle GED) = 126^\circ$).

Point G is on AB, point E is on CD.

To Find:

The measures of angles $\angle AGE$, $\angle GEF$, and $\angle FGE$.

Solution:

We are given that $AB \parallel CD$ and $GE$ is a transversal line intersecting AB at G and CD at E.

The angles $\angle AGE$ and $\angle GED$ are consecutive interior angles formed by the transversal GE with the parallel lines AB and CD.

By the property of consecutive interior angles, if a transversal intersects two parallel lines, the sum of the measures of consecutive interior angles on the same side of the transversal is $180^\circ$ (they are supplementary).

$m(\angle AGE) + m(\angle GED) = 180^\circ$

Substitute the given value $m(\angle GED) = 126^\circ$:

$m(\angle AGE) + 126^\circ = 180^\circ$

Subtract $126^\circ$ from both sides to find $m(\angle AGE)$:

$m(\angle AGE) = 180^\circ - 126^\circ$

$m(\angle AGE) = 54^\circ$

We are given that $EF \perp CD$ at E. This means the line containing EF is perpendicular to the line CD at point E.

Let the ray from E perpendicular to CD, on the same side of CD as G, be denoted by ray EY. Then $\angle DEY = 90^\circ$.

We are given $\angle GED = 126^\circ$. The ray EG forms an angle of $126^\circ$ with ray ED.

Since $90^\circ < 126^\circ$, the ray EY must lie in the interior of $\angle GED$.

Therefore, $\angle GED$ can be expressed as the sum of $\angle GEY$ and $\angle YED$:

$m(\angle GED) = m(\angle GEY) + m(\angle YED)$

Substitute the known values:

$126^\circ = m(\angle GEY) + 90^\circ$

Subtract $90^\circ$ from both sides to find $m(\angle GEY)$:

$m(\angle GEY) = 126^\circ - 90^\circ$

$m(\angle GEY) = 36^\circ$

The point F lies on the line containing ray EY (since EF $\perp$ CD at E). Assuming F is on the ray EY, then $\angle GEF$ is the same as $\angle GEY$.

$m(\angle GEF) = 36^\circ$

Now we need to find $\angle FGE$. G is on AB, E is on CD, and F is on the line perpendicular to CD at E.

Since the line containing EF is perpendicular to CD and $AB \parallel CD$, the line containing EF is also perpendicular to AB.

Let H be the point where the line containing EF (the line EY) intersects AB. Then $EH \perp AB$ at H.

H is a point on AB, and G is also a point on AB. E is on CD. Consider the triangle $\triangle GHE$.

In $\triangle GHE$, $\angle GHE = 90^\circ$ (since $EH \perp AB$).

The angles in $\triangle GHE$ are $\angle HGE$, $\angle GHE$, and $\angle HEG$.

We know $\angle HEG$. Ray EG makes angle $\angle HEG$ with ray EH. Ray EH is the same as ray EY from step 2. So $\angle HEG = \angle GEY = 36^\circ$.

By the Angle Sum Property of a triangle, the sum of the interior angles of $\triangle GHE$ is $180^\circ$:

$m(\angle HGE) + m(\angle GHE) + m(\angle HEG) = 180^\circ$

Substitute the known values:

$m(\angle HGE) + 90^\circ + 36^\circ = 180^\circ$

$m(\angle HGE) + 126^\circ = 180^\circ$

Subtract $126^\circ$ from both sides to find $m(\angle HGE)$:

$m(\angle HGE) = 180^\circ - 126^\circ$

$m(\angle HGE) = 54^\circ$

The point F lies on the line containing EH. The point H lies on the line AB. The point G lies on the line AB.

If F is on the ray EH, then ray GF is the same as ray GH (as G and H are on line AB, and F is on the line EH). The angle $\angle FGE$ is the angle between ray GF and ray GE, which is the same as the angle between ray GH and ray GE, i.e., $\angle HGE$.

$m(\angle FGE) = m(\angle HGE) = 54^\circ$

Summary of Results:

$m(\angle AGE) = 54^\circ$

$m(\angle GEF) = 36^\circ$

$m(\angle FGE) = 54^\circ$

Given:

Two lines $l$ and $m$ are intersected by a transversal $t$ at points P and Q, respectively.

Let $\angle APQ$ be a corresponding angle formed by line $l$ and transversal $t$ at P, and $\angle PQR$ be the corresponding angle formed by line $m$ and transversal $t$ at Q (where A is a point on $l$, R is a point on $m$, and A and R are on the same side of the transversal $t$). So, $\angle APQ$ and $\angle PQR$ are corresponding angles.

Let PX be the bisector of $\angle APQ$ and QY be the bisector of $\angle PQR$.

We are given that the bisectors PX and QY are parallel, i.e., $PX \parallel QY$.

To Prove:

The lines $l$ and $m$ are parallel, i.e., $l \parallel m$.

Diagram:

Draw two lines $l$ and $m$ (not assumed parallel) intersected by a transversal $t$. Mark the intersection points P on $l$ and Q on $m$. Identify a pair of corresponding angles, for example, the upper-left angle at P ($\angle APQ$) and the upper-left angle at Q ($\angle PQR$). Draw the angle bisector PX for $\angle APQ$ and the angle bisector QY for $\angle PQR$. Indicate that PX is parallel to QY.

Proof:

Since PX is the bisector of $\angle APQ$, by the definition of an angle bisector, it divides the angle into two equal halves.

$m(\angle XPQ) = \frac{1}{2} m(\angle APQ)$

Similarly, since QY is the bisector of $\angle PQR$, it divides the angle into two equal halves.

$m(\angle YQP) = \frac{1}{2} m(\angle PQR)$

We are given that the bisectors PX and QY are parallel ($PX \parallel QY$), and $t$ is a transversal intersecting them at P and Q.

Consider the angles $\angle XPQ$ and $\angle YQP$ formed by the parallel lines PX, QY and the transversal PQ (which is part of line $t$). These angles are alternate interior angles with respect to the lines PX and QY and transversal PQ.

Since PX $\parallel$ QY, the alternate interior angles are equal:

Substitute the expressions for $m(\angle XPQ)$ and $m(\angle YQP)$ from the bisector definitions:

$\frac{1}{2} m(\angle APQ) = \frac{1}{2} m(\angle PQR)$

Multiply both sides of the equation by 2:

$m(\angle APQ) = m(\angle PQR)$

Now, consider lines $l$ and $m$ intersected by the transversal $t$. We have shown that $\angle APQ$ and $\angle PQR$, which are a pair of corresponding angles, are equal in measure.

By the Converse of the Corresponding Angles Axiom, if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.

Since $m(\angle APQ) = m(\angle PQR)$, we can conclude that $l \parallel m$.

Therefore, if a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then the two lines are parallel.

Hence, Proved.

Given:

Lines PQ and RS intersect at point O.

The ratio of $\angle$POR to $\angle$ROQ is $5:7$ ($m(\angle \text{POR}) : m(\angle \text{ROQ}) = 5 : 7$).

To Find:

The measure of all four angles formed by the intersection: $\angle$POR, $\angle$ROQ, $\angle$QOS, and $\angle$SOP.

Solution:

When two lines PQ and RS intersect at O, the angles $\angle$POR and $\angle$ROQ form a linear pair on the line RS.

By the Linear Pair Axiom, the sum of the measures of angles in a linear pair is $180^\circ$.

The ratio of $m(\angle \text{POR})$ to $m(\angle \text{ROQ})$ is given as $5:7$.

Let the common ratio constant be $k$. Then we can write the measures of the angles as:

$m(\angle \text{POR}) = 5k$

$m(\angle \text{ROQ}) = 7k$

Substitute these expressions into the linear pair equation:

$5k + 7k = 180^\circ$

Combine the terms on the left side:

$12k = 180^\circ$

Solve for $k$ by dividing both sides by 12:

$k = \frac{180^\circ}{12}$

$k = 15^\circ$

Now, substitute the value of $k$ back into the expressions for the angles to find their measures:

$m(\angle \text{POR}) = 5k = 5 \times 15^\circ = 75^\circ$

$m(\angle \text{ROQ}) = 7k = 7 \times 15^\circ = 105^\circ$

Next, we find the measures of the other two angles using the property of vertically opposite angles.

$\angle$POR and $\angle$QOS are vertically opposite angles. Vertically opposite angles are equal.

So, $m(\angle \text{QOS}) = 75^\circ$.

Similarly, $\angle$ROQ and $\angle$SOP are vertically opposite angles. Vertically opposite angles are equal.

So, $m(\angle \text{SOP}) = 105^\circ$.

Alternatively, after finding $m(\angle \text{POR}) = 75^\circ$, we could find $m(\angle \text{SOP})$. $\angle$POR and $\angle$SOP form a linear pair on line PQ.

$m(\angle \text{POR}) + m(\angle \angle \text{SOP}) = 180^\circ$ (Linear Pair Axiom)

$75^\circ + m(\angle \text{SOP}) = 180^\circ$

$m(\angle \text{SOP}) = 180^\circ - 75^\circ = 105^\circ$. This matches the vertically opposite method.

Similarly, $m(\angle \text{ROQ}) + m(\angle \text{QOS}) = 180^\circ$ (Linear Pair Axiom).

$105^\circ + m(\angle \text{QOS}) = 180^\circ$

$m(\angle \text{QOS}) = 180^\circ - 105^\circ = 75^\circ$. This also matches the vertically opposite method.

The measures of the four angles are:

$\angle$POR = $75^\circ$

$\angle$ROQ = $105^\circ$

$\angle$QOS = $75^\circ$

$\angle$SOP = $105^\circ$

Given:

A triangle ABC. Let the interior angles be $\angle A$, $\angle B$, and $\angle C$ at vertices A, B, and C respectively.

To Prove:

The sum of the exterior angles of triangle ABC, taken in order, is $360^\circ$.

Diagram:

Imagine a triangle ABC. Extend the side BC to a point D, side CA to a point E, and side AB to a point F. The exterior angles taken in order are $\angle ACD$, $\angle BAE$, and $\angle CBF$. These angles are adjacent to the interior angles $\angle C$, $\angle A$, and $\angle B$ respectively.

Proof:

Consider the exterior angle at vertex C, which is $\angle ACD$. By the Exterior Angle Property of a triangle, the measure of an exterior angle is equal to the sum of the measures of its two interior opposite angles.

Similarly, consider the exterior angle at vertex A, which is $\angle BAE$. The interior opposite angles are $\angle B$ and $\angle C$.

Finally, consider the exterior angle at vertex B, which is $\angle CBF$. The interior opposite angles are $\angle C$ and $\angle A$.

Now, let's find the sum of these three exterior angles:

Sum of exterior angles $= m(\angle \text{ACD}) + m(\angle \text{BAE}) + m(\angle \text{CBF})$

Substitute the expressions from the Exterior Angle Property:

Sum $= (m(\angle A) + m(\angle B)) + (m(\angle B) + m(\angle C)) + (m(\angle C) + m(\angle A))$

Rearrange and group the terms:

Sum $= m(\angle A) + m(\angle A) + m(\angle B) + m(\angle B) + m(\angle C) + m(\angle C)$

Sum $= 2 \times m(\angle A) + 2 \times m(\angle B) + 2 \times m(\angle C)$

Sum $= 2 (m(\angle A) + m(\angle B) + m(\angle C))$

By the Angle Sum Property of a triangle, the sum of the interior angles of a triangle is $180^\circ$.

Substitute this value into the expression for the sum of exterior angles:

Sum $= 2 (180^\circ)$

Sum $= 360^\circ$

Therefore, the sum of the exterior angles of a triangle taken in order is $360^\circ$.

Hence, Proved.

Alternative Proof using Linear Pairs:

Each interior angle and its adjacent exterior angle form a linear pair, so their sum is $180^\circ$.

$m(\angle A) + m(\angle \text{Exterior at A}) = 180^\circ$

$m(\angle B) + m(\angle \text{Exterior at B}) = 180^\circ$

$m(\angle C) + m(\angle \text{Exterior at C}) = 180^\circ$

Summing these three equations:

$(m(\angle A) + m(\angle B) + m(\angle C)) + (m(\angle \text{Exterior at A}) + m(\angle \text{Exterior at B}) + m(\angle \text{Exterior at C})) = 180^\circ + 180^\circ + 180^\circ$

We know $m(\angle A) + m(\angle B) + m(\angle C) = 180^\circ$ (Angle Sum Property).

$180^\circ + (\text{Sum of exterior angles}) = 540^\circ$

Subtract $180^\circ$ from both sides:

Sum of exterior angles $= 540^\circ - 180^\circ$

Sum of exterior angles $= 360^\circ$

This method also proves the theorem using the Angle Sum Property and the Linear Pair Axiom.

Given:

Line $PQ$ is parallel to line $ST$ ($PQ \parallel ST$).

The measure of angle $\angle PQR$ is $110^\circ$ ($m(\angle PQR) = 110^\circ$).

The measure of angle $\angle RST$ is $130^\circ$ ($m(\angle RST) = 130^\circ$).

To Find:

The measure of angle $\angle QRS$ ($m(\angle QRS)$).

Construction:

Draw a line XRY passing through point R such that XRY is parallel to PQ and ST. Let X be a point on the line to the left of R, and Y be a point on the line to the right of R.

Thus, $XRY \parallel PQ$ and $XRY \parallel ST$.

Diagram:

Imagine two parallel horizontal lines, PQ above and ST below. A point R is between them. Segment QR goes from Q on PQ to R. Segment RS goes from R to S on ST. Draw a horizontal line XRY through R with X to the left and Y to the right.

Solution:

Consider the parallel lines PQ and XRY intersected by the transversal QR.

Angles $\angle PQR$ and $\angle QRX$ are consecutive interior angles (interior angles on the same side of the transversal).

By the property of consecutive interior angles, their sum is $180^\circ$.

Substitute the given value $m(\angle PQR) = 110^\circ$:

$110^\circ + m(\angle QRX) = 180^\circ$

$m(\angle QRX) = 180^\circ - 110^\circ$

$m(\angle QRX) = 70^\circ$

So, the angle between ray RQ and ray RX is $70^\circ$.

Now, consider the parallel lines ST and XRY intersected by the transversal RS.

Angles $\angle RST$ and $\angle SRY$ are consecutive interior angles (interior angles on the same side of the transversal).

By the property of consecutive interior angles, their sum is $180^\circ$.

Substitute the given value $m(\angle RST) = 130^\circ$:

$130^\circ + m(\angle SRY) = 180^\circ$

$m(\angle SRY) = 180^\circ - 130^\circ$

$m(\angle SRY) = 50^\circ$

So, the angle between ray RS and ray RY is $50^\circ$.

The line XRY is a straight line, so $\angle XRY = 180^\circ$. Ray RX and ray RY are opposite rays.

The angle $\angle QRS$ is the angle formed by the rays RQ and RS. Ray RQ and ray RS are on the same side of the line XRY (specifically, they are between the parallel lines PQ and ST).

Consider ray RX as the common arm. The angle between ray RX and ray RQ is $m(\angle QRX) = 70^\circ$.

The angle between ray RX and ray RS is $m(\angle SRX)$. The angles $\angle SRX$ and $\angle SRY$ form a linear pair on the line XRY.

Substitute $m(\angle SRY) = 50^\circ$:

$m(\angle SRX) + 50^\circ = 180^\circ$

$m(\angle SRX) = 180^\circ - 50^\circ$

$m(\angle SRX) = 130^\circ$

So, the angle between ray RX and ray RS is $130^\circ$.

Now, consider the angles around R with ray RX as a common arm: $\angle QRX$ (between RX and RQ) and $\angle SRX$ (between RX and RS). The angle $\angle QRS$ is the angle between RQ and RS.

We have $m(\angle QRX) = 70^\circ$ and $m(\angle SRX) = 130^\circ$.

Since $70^\circ < 130^\circ$, the ray RQ lies in the interior of the angle $\angle SRX$ (the angle between ray RX and ray RS).

Therefore, by angle addition, the measure of the larger angle $\angle SRX$ is the sum of the measures of the two adjacent angles $\angle SRQ$ (or $\angle QRS$) and $\angle QRX$.

Substitute the known values:

$130^\circ = 70^\circ + m(\angle QRS)$

Subtract $70^\circ$ from both sides to find $m(\angle QRS)$:

$m(\angle QRS) = 130^\circ - 70^\circ$

$m(\angle QRS) = 60^\circ$

Thus, the measure of angle $\angle QRS$ is $60^\circ$.

Given:

In $\triangle XYZ$, $m(\angle X) = 62^\circ$ and $m(\angle XYZ) = 54^\circ$.

YO is the bisector of $\angle XYZ$.

ZO is the bisector of $\angle XZY$.

To Find:

The measures of $\angle OZY$ and $\angle YOZ$.

Solution:

In $\triangle XYZ$, the sum of the interior angles is $180^\circ$ by the Angle Sum Property.

$m(\angle X) + m(\angle XYZ) + m(\angle XZY) = 180^\circ$

Substitute the given values of $m(\angle X)$ and $m(\angle XYZ)$:

$62^\circ + 54^\circ + m(\angle XZY) = 180^\circ$

Combine the known angles:

$116^\circ + m(\angle XZY) = 180^\circ$

Subtract $116^\circ$ from both sides to find $m(\angle XZY)$:

$m(\angle XZY) = 180^\circ - 116^\circ$

$m(\angle XZY) = 64^\circ$

So, the measure of angle $\angle XZY$ is $64^\circ$.

Now, consider the angle bisectors.

YO is the bisector of $\angle XYZ$. This means YO divides $\angle XYZ$ into two equal angles: $\angle XY O$ and $\angle OYZ$.

$m(\angle OYZ) = \frac{1}{2} m(\angle XYZ)$

$m(\angle OYZ) = \frac{1}{2} \times 54^\circ$

$m(\angle OYZ) = 27^\circ$

ZO is the bisector of $\angle XZY$. This means ZO divides $\angle XZY$ into two equal angles: $\angle XZO$ and $\angle OZY$.

$m(\angle OZY) = \frac{1}{2} m(\angle XZY)$

We found $m(\angle XZY) = 64^\circ$.

$m(\angle OZY) = \frac{1}{2} \times 64^\circ$

$m(\angle OZY) = 32^\circ$

This is one of the angles we needed to find.

Finally, consider $\triangle OYZ$. The sum of the interior angles of $\triangle OYZ$ is $180^\circ$ by the Angle Sum Property.

$m(\angle OYZ) + m(\angle OZY) + m(\angle YOZ) = 180^\circ$

Substitute the values we found for $m(\angle OYZ)$ and $m(\angle OZY)$:

$27^\circ + 32^\circ + m(\angle YOZ) = 180^\circ$

Combine the known angles:

$59^\circ + m(\angle YOZ) = 180^\circ$

Subtract $59^\circ$ from both sides to find $m(\angle YOZ)$:

$m(\angle YOZ) = 180^\circ - 59^\circ$

$m(\angle YOZ) = 121^\circ$

This is the other angle we needed to find.

Thus, the measure of $\angle OZY$ is $32^\circ$ and the measure of $\angle YOZ$ is $121^\circ$.

Given:

Two lines $l$ and $m$ are intersected by a transversal $t$ at points P and Q, respectively.

A pair of alternate interior angles formed are equal. Let's assume, using the standard numbering for a transversal cutting two lines, that $\angle 4$ and $\angle 6$ are alternate interior angles, and $m(\angle 4) = m(\angle 6)$.

To Prove:

The lines $l$ and $m$ are parallel, i.e., $l \parallel m$.

Diagram:

Imagine two lines, labelled $l$ and $m$, drawn horizontally (they are not necessarily parallel at this stage). Draw a line, labelled $t$, intersecting $l$ at point P and $m$ at point Q. Label the angles formed around P and Q. Let the angles at P, starting from the top-left and going clockwise, be $\angle 1, \angle 2, \angle 3, \angle 4$. Let the angles at Q, starting from the top-left and going clockwise, be $\angle 5, \angle 6, \angle 7, \angle 8$. The alternate interior angles are $\angle 4$ (at P, bottom-left interior) and $\angle 6$ (at Q, top-right interior), and $\angle 3$ (at P, bottom-right interior) and $\angle 5$ (at Q, top-left interior). We are given that $m(\angle 4) = m(\angle 6)$ (or $m(\angle 3) = m(\angle 5)$).

Explanation of Proof:

We are given that the alternate interior angles $\angle 4$ and $\angle 6$ are equal in measure ($m(\angle 4) = m(\angle 6)$).

Consider the angles at the intersection point P on line $l$. Angles $\angle 4$ and $\angle 2$ are vertically opposite angles.

We know that vertically opposite angles are always equal.

Now we have two equalities: $m(\angle 4) = m(\angle 6)$ (Given) and $m(\angle 4) = m(\angle 2)$.

By the transitive property of equality, if two quantities are equal to the same third quantity, they are equal to each other.

Thus, $m(\angle 2) = m(\angle 6)$.

Look at the angles $\angle 2$ and $\angle 6$ in the diagram. $\angle 2$ is at P (top-right exterior) and $\angle 6$ is at Q (top-right interior). These are a pair of corresponding angles.

We have shown that a pair of corresponding angles ($m(\angle 2)$ and $m(\angle 6)$) are equal in measure.

According to the Converse of the Corresponding Angles Axiom (which states that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel), we can conclude that the lines $l$ and $m$ must be parallel.

Therefore, if a transversal intersects two lines such that a pair of alternate interior angles is equal, the two lines are parallel.

Given:

Lines AB, CD, and EF are three concurrent lines intersecting at point O.

The measures of some angles are given as: $m(\angle AOF) = 2y$, $m(\angle FOB) = 5y$, $m(\angle COE) = 3y$.

Angles around O are labelled with measures $x, 2y, 5y, 3y, z$ (among others).

To Find:

The values of $x, y,$ and $z$.

Solution:

When three lines intersect at a single point O, they form six angles around that point. These six angles form three pairs of vertically opposite angles. Vertically opposite angles are equal in measure.

The sum of the measures of all angles around the point of intersection is $360^\circ$.

From the given angle measures and the property of vertically opposite angles:

$m(\angle AOF) = 2y$. Its vertically opposite angle is $\angle EOD$, so $m(\angle EOD) = 2y$.

$m(\angle FOB) = 5y$. Its vertically opposite angle is $\angle AOE$, so $m(\angle AOE) = 5y$.

$m(\angle COE) = 3y$. Its vertically opposite angle is $\angle DOF$, so $m(\angle DOF) = 3y$.

The six angles formed around O are $\angle AOF, \angle FOB, \angle BOC, \angle COE, \angle EOD, \angle DOA$. Their measures are $2y, 5y, m(\angle BOC), 3y, m(\angle EOD), m(\angle DOA)$.

Using the vertically opposite angle relationships, we have:

$m(\angle EOD) = 2y$ (vertically opposite to $\angle AOF$)

$m(\angle DOA) = m(\angle FOB) = 5y$ (vertically opposite to $\angle FOB$)

$m(\angle BOC) = m(\angle DOF) = 3y$ (vertically opposite to $\angle DOF$, and $\angle DOF$ is vertically opposite to $\angle COE$)

The measures of the six angles around O are $2y, 5y, 3y, 3y, 2y, 5y$.

The sum of these measures is $360^\circ$:

$2y + 5y + 3y + 3y + 2y + 5y = 360^\circ$

$(2 + 5 + 3 + 3 + 2 + 5)y = 360^\circ$

$20y = 360^\circ$

Solving for $y$:

$y = \frac{360^\circ}{20}$

$y = 18^\circ$

Now we use the derived value of $y$ to find the measures of the angles:

The measures of the three pairs of vertically opposite angles are $2y = 2(18^\circ) = 36^\circ$, $5y = 5(18^\circ) = 90^\circ$, and $3y = 3(18^\circ) = 54^\circ$.

So the six angles around O have measures $36^\circ, 36^\circ, 54^\circ, 54^\circ, 90^\circ, 90^\circ$.

The angles are labelled $x, 2y, 5y, 3y, z$. Substituting $y=18^\circ$, these labels represent measures $x, 36^\circ, 90^\circ, 54^\circ, z$.

Based on the structure of the problem and typical angle labelling in such diagrams, the labels $x$ and $z$ correspond to the measures of the other angles.

In our derivation based on vertical angles, the six angles around O (like $\angle AOF, \angle FOB, \angle BOC, \angle COE, \angle EOD, \angle DOA$) have measures $2y, 5y, 3y, 3y, 2y, 5y$ in some order.

Mapping the labels to specific angles based on the order mentioned in the problem description (often implies adjacent angles in the diagram) and vertical angle properties:

Let $\angle AOF = 2y$, $\angle FOB = 5y$, $\angle BOC = x$, $\angle COE = 3y$, $\angle EOD = z$, $\angle DOA = w$.

Vertical opposite pairs are $(\angle AOF, \angle EOD)$, $(\angle FOB, \angle DOA)$, $(\angle BOC, \angle DOF)$.

This implies: $m(\angle AOF) = m(\angle EOD) \implies 2y = z$.

$m(\angle FOB) = m(\angle DOA) \implies 5y = w$.

$m(\angle BOC) = m(\angle DOF) \implies x = m(\angle DOF)$.

Also, $m(\angle DOF) = m(\angle COE)$ (vertically opposite) $\implies m(\angle DOF) = 3y$.

From $x = m(\angle DOF)$, we get $x = 3y$.

Using $y = 18^\circ$:

$x = 3y = 3 \times 18^\circ = 54^\circ$

$z = 2y = 2 \times 18^\circ = 36^\circ$

Let's check these values:

$y = 18^\circ$

$x = 54^\circ$

$z = 36^\circ$

The measures of the angles based on this assignment are:

$\angle AOF = 2y = 36^\circ$

$\angle FOB = 5y = 90^\circ$

$\angle BOC = x = 54^\circ$

$\angle COE = 3y = 54^\circ$

$\angle EOD = z = 36^\circ$

$\angle DOA = 5y = 90^\circ$ (since $m(\angle DOA) = 5y$ from vertical opposite to $\angle FOB$)

$m(\angle DOF) = 3y = 54^\circ$ (from vertical opposite to $\angle COE$)

These measures satisfy the vertical angle relationships:

$m(\angle AOF) = 36^\circ$ and $m(\angle EOD) = 36^\circ$.

$m(\angle FOB) = 90^\circ$ and $m(\angle DOA) = 90^\circ$.

$m(\angle BOC) = 54^\circ$ and $m(\angle DOF) = 54^\circ$.

The sum of the measures around O is $36^\circ + 90^\circ + 54^\circ + 54^\circ + 36^\circ + 90^\circ = 360^\circ$, which is consistent.

Note: These values satisfy the angle sum around the point and the vertical angle properties based on a plausible interpretation of the labels and angle relationships. However, the resulting angle values do not form perfect straight lines (e.g., $\angle AOF + \angle FOB = 36^\circ + 90^\circ = 126^\circ \neq 180^\circ$), which indicates a potential inconsistency within the problem statement itself, assuming AB, CD, and EF are precisely straight lines as drawn in Euclidean geometry diagrams.

Assuming the calculation method based on the provided expressions and vertical angles is the intended approach to find the values of $x, y, z$:

The values are:

$x = 54^\circ$

$y = 18^\circ$

$z = 36^\circ$