Chapter 5 Lines and Angles

Solution:

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

The complement of an angle with measure $x^\circ$ is $(90 - x)^\circ$.

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(i) The given angle is $20^\circ$.

The complement of $20^\circ$ is $(90 - 20)^\circ = 70^\circ$.

$\text{Complement} = 90^\circ - 20^\circ = 70^\circ$

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(ii) The given angle is $63^\circ$.

The complement of $63^\circ$ is $(90 - 63)^\circ = 27^\circ$.

$\text{Complement} = 90^\circ - 63^\circ = 27^\circ$

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(iii) The given angle is $57^\circ$.

The complement of $57^\circ$ is $(90 - 57)^\circ = 33^\circ$.

$\text{Complement} = 90^\circ - 57^\circ = 33^\circ$

Solution:

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

The supplement of an angle with measure $x^\circ$ is $(180 - x)^\circ$.

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(i) The given angle is $105^\circ$.

The supplement of $105^\circ$ is $(180 - 105)^\circ = 75^\circ$.

$\text{Supplement} = 180^\circ - 105^\circ = 75^\circ$

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(ii) The given angle is $87^\circ$.

The supplement of $87^\circ$ is $(180 - 87)^\circ = 93^\circ$.

$\text{Supplement} = 180^\circ - 87^\circ = 93^\circ$

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(iii) The given angle is $154^\circ$.

The supplement of $154^\circ$ is $(180 - 154)^\circ = 26^\circ$.

$\text{Supplement} = 180^\circ - 154^\circ = 26^\circ$

Solution:

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

Two angles are supplementary if their sum is $180^\circ$.

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(i) 65º, 115º

Sum of angles $= 65^\circ + 115^\circ = 180^\circ$.

Since the sum is $180^\circ$, the pair of angles is supplementary.

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(ii) 63º, 27º

Sum of angles $= 63^\circ + 27^\circ = 90^\circ$.

Since the sum is $90^\circ$, the pair of angles is complementary.

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(iii) 112º, 68º

Sum of angles $= 112^\circ + 68^\circ = 180^\circ$.

Since the sum is $180^\circ$, the pair of angles is supplementary.

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(iv) 130º, 50º

Sum of angles $= 130^\circ + 50^\circ = 180^\circ$.

Since the sum is $180^\circ$, the pair of angles is supplementary.

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(v) 45º, 45º

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

Since the sum is $90^\circ$, the pair of angles is complementary.

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(vi) 80º, 10º

Sum of angles $= 80^\circ + 10^\circ = 90^\circ$.

Since the sum is $90^\circ$, the pair of angles is complementary.

Solution:

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To Find:

The angle which is equal to its complement.

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Let:

Let the angle be $x^\circ$.

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Concept:

The complement of an angle $x^\circ$ is $(90 - x)^\circ$.

According to the problem, the angle is equal to its complement.

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Equation:

So, we can write the equation:

$x = 90 - x$

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Solve:

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

$x + x = 90 - x + x$

$2x = 90$

Now, divide both sides by 2:

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

$x = 45$

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The angle is $45^\circ$. Its complement is $(90 - 45)^\circ = 45^\circ$. The angle is indeed equal to its complement.

Answer:

The angle which is equal to its complement is $\mathbf{45^\circ}$.

Solution:

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To Find:

The angle which is equal to its supplement.

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Let:

Let the angle be $x^\circ$.

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Concept:

The supplement of an angle $x^\circ$ is $(180 - x)^\circ$.

According to the problem, the angle is equal to its supplement.

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Equation:

So, we can write the equation:

$x = 180 - x$

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Solve:

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

$x + x = 180 - x + x$

$2x = 180$

Now, divide both sides by 2:

$\frac{2x}{2} = \frac{180}{2}$

$x = 90$

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The angle is $90^\circ$. Its supplement is $(180 - 90)^\circ = 90^\circ$. The angle is indeed equal to its supplement.

Answer:

The angle which is equal to its supplement is $\mathbf{90^\circ}$.

Solution:

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Given that $\angle 1$ and $\angle 2$ are supplementary angles.

This means that the sum of their measures is $180^\circ$.

$\angle 1 + \angle 2 = 180^\circ$

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If $\angle 1$ is decreased, for the sum of the two angles to remain $180^\circ$, $\angle 2$ must change.

Let's consider an example.

Suppose $\angle 1 = 100^\circ$ and $\angle 2 = 80^\circ$. Their sum is $100^\circ + 80^\circ = 180^\circ$, so they are supplementary.

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Now, let's decrease $\angle 1$ by $10^\circ$. The new $\angle 1$ becomes $100^\circ - 10^\circ = 90^\circ$. The amount of decrease is $10^\circ$.

For the angles to remain supplementary, the new $\angle 1$ and the new $\angle 2$ must still add up to $180^\circ$.

$90^\circ + \text{New } \angle 2 = 180^\circ$.

This means the New $\angle 2 = 180^\circ - 90^\circ = 90^\circ$.

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The original $\angle 2$ was $80^\circ$, and the new $\angle 2$ is $90^\circ$.

The change in $\angle 2$ is $90^\circ - 80^\circ = 10^\circ$. This is an increase.

We decreased $\angle 1$ by $10^\circ$ and $\angle 2$ increased by $10^\circ$. The amount of decrease in $\angle 1$ is equal to the amount of increase in $\angle 2$.

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Therefore, if $\angle 1$ is decreased by some amount, $\angle 2$ must be increased by the same amount so that both the angles still remain supplementary.

Solution:

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

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(i) acute?

An acute angle has a measure less than $90^\circ$.

If we take two acute angles, the measure of each angle is less than $90^\circ$.

The sum of two acute angles will be less than $90^\circ + 90^\circ$.

Sum of two acute angles $< 180^\circ$.

Since the sum is less than $180^\circ$, two acute angles cannot be supplementary.

Answer: No

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(ii) obtuse?

An obtuse angle has a measure greater than $90^\circ$.

If we take two obtuse angles, the measure of each angle is greater than $90^\circ$.

The sum of two obtuse angles will be greater than $90^\circ + 90^\circ$.

Sum of two obtuse angles $> 180^\circ$.

Since the sum is greater than $180^\circ$, two obtuse angles cannot be supplementary.

Answer: No

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(iii) right?

A right angle has a measure exactly equal to $90^\circ$.

If we take two right angles, the measure of each angle is exactly $90^\circ$.

The sum of two right angles is $90^\circ + 90^\circ$.

Sum of two right angles $= 180^\circ$.

Since the sum is exactly $180^\circ$, two right angles can be supplementary.

Answer: Yes

Solution:

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

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Let the given angle be Angle 1 and its complementary angle be Angle 2.

We know that:

Angle 1 + Angle 2 = $90^\circ$

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We are given that Angle 1 is greater than $45^\circ$.

Angle 1 $> 45^\circ$

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Since Angle 1 is greater than $45^\circ$, it means that Angle 1 takes up more than half of the total sum of $90^\circ$.

For example, if Angle 1 was exactly $45^\circ$, then Angle 2 would be $90^\circ - 45^\circ = 45^\circ$.

If Angle 1 is greater than $45^\circ$ (like $50^\circ$, $60^\circ$, etc.), then Angle 2 must be smaller than $45^\circ$ to make the sum $90^\circ$.

Let's check with an example:

Suppose Angle 1 = $50^\circ$. This is greater than $45^\circ$.

Angle 1 + Angle 2 = $90^\circ$

$50^\circ$ + Angle 2 = $90^\circ$

Angle 2 = $90^\circ - 50^\circ$

Angle 2 = $40^\circ$

$40^\circ$ is less than $45^\circ$.

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So, if an angle is greater than $45^\circ$, its complementary angle must be the remaining part to reach $90^\circ$. This remaining part will be less than $45^\circ$.

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Therefore, if an angle is greater than $45^\circ$, its complementary angle is less than 45º.

Solution:

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Let's fill in the blanks based on the definitions of complementary and supplementary angles, and the property of adjacent angles.

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(i) If two angles are complementary, then the sum of their measures is $\underline{\mathbf{90^\circ}}$.

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(ii) If two angles are supplementary, then the sum of their measures is $\underline{\mathbf{180^\circ}}$.

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(iii) If two adjacent angles are supplementary, they form a $\underline{\text{linear pair}}$.

Solution:

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Lines AC and BD intersect at point O.

$\angle$AOD is an obtuse angle.

Ray OE is perpendicular to line BD, and it is drawn between $\angle$AOD.

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(i) Obtuse vertically opposite angles

Vertically opposite angles are formed by two intersecting lines. They are opposite to each other at the vertex and are equal.

The intersecting lines are AC and BD. The pairs of vertically opposite angles are ($\angle$AOD, $\angle$BOC) and ($\angle$AOB, $\angle$DOC).

We are given that $\angle$AOD is obtuse. Since $\angle$BOC is vertically opposite to $\angle$AOD, $\angle$BOC is also obtuse.

$\angle$AOB and $\angle$DOC are vertically opposite. Since $\angle$AOD and $\angle$AOB form a linear pair on line BD ($\angle$AOD + $\angle$AOB = $180^\circ$), and $\angle$AOD is obtuse ($> 90^\circ$), $\angle$AOB must be acute ($< 90^\circ$). Similarly, $\angle$DOC is acute.

Therefore, the obtuse vertically opposite angles are ($\angle$AOD, $\angle$BOC).

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(ii) Adjacent complementary angles

Adjacent angles share a common vertex and a common arm. Complementary angles are two angles whose sum is $90^\circ$.

We are given that ray OE is perpendicular to line BD. This means that the angle formed by OE and BD is $90^\circ$. Thus, $\angle$EOB = $90^\circ$.

The angle $\angle$EOB is formed by the adjacent angles $\angle$EOA and $\angle$AOB. They share the common vertex O and the common arm OA.

The sum of these adjacent angles is $\angle$EOA + $\angle$AOB = $\angle$EOB.

$\angle$EOA + $\angle$AOB = $90^\circ$

Since their sum is $90^\circ$, $\angle$EOA and $\angle$AOB are complementary.

Therefore, the adjacent complementary angles are ($\angle$EOA, $\angle$AOB).

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(iii) Equal supplementary angles

Supplementary angles are two angles whose sum is $180^\circ$. Equal supplementary angles must each measure $180^\circ / 2 = 90^\circ$.

We know that OE is perpendicular to line BD. This implies $\angle$EOB = $90^\circ$ and $\angle$EOD = $90^\circ$.

These two angles, $\angle$EOB and $\angle$EOD, share the common vertex O and the common arm OE. Their non-common arms OB and OD form the straight line BD. Thus, they are adjacent and form a linear pair.

Since $\angle$EOB = $\angle$EOD = $90^\circ$, they are equal.

Their sum is $\angle$EOB + $\angle$EOD = $90^\circ + 90^\circ = 180^\circ$, so they are supplementary.

Therefore, the equal supplementary angles are ($\angle$EOB, $\angle$EOD).

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(iv) Unequal supplementary angles

Supplementary angles are two angles whose sum is $180^\circ$. Unequal supplementary angles add up to $180^\circ$ but have different measures.

Consider the angles $\angle$EOA and $\angle$EOC. They are adjacent angles sharing the common vertex O and the common arm OE. Their non-common arms OA and OC form the straight line AC.

Thus, $\angle$EOA and $\angle$EOC form a linear pair, so their sum is $180^\circ$. They are supplementary.

Since, $\angle$EOA is acute (from figure/deduction if $\angle$AOD is obtuse) and $\angle$EOC is obtuse. Therefore, $\angle$EOA $\neq \angle$EOC.

Thus, the unequal supplementary angles are ($\angle$EOA, $\angle$EOC).

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(v) Adjacent angles that do not form a linear pair

Adjacent angles share a common vertex and a common arm. A linear pair is a special case of adjacent angles where the non-common arms form a straight line, and their sum is $180^\circ$. We need adjacent angles whose non-common arms do not form a straight line (i.e., their sum is not $180^\circ$).

Let's look at the given pairs:

($\angle$AOB, $\angle$AOE): These angles are adjacent (common vertex O, common arm OA). Their sum is $\angle$AOB + $\angle$AOE = $\angle$EOB. We know $\angle$EOB = $90^\circ$. Since $90^\circ \neq 180^\circ$, they do not form a linear pair.

($\angle$AOE, $\angle$EOD): These angles are adjacent (common vertex O, common arm OE). Their sum is $\angle$AOE + $\angle$EOD = $\angle$AOD. We are given that $\angle$AOD is obtuse. Since an obtuse angle is $> 90^\circ$ and not $180^\circ$ (unless it's a straight angle, which $\angle$AOD is not, as AC is a straight line), $\angle$AOD is not $180^\circ$. So, they do not form a linear pair.

($\angle$EOD, $\angle$COD): These angles are adjacent (common vertex O, common arm OD). Their sum is $\angle$EOD + $\angle$COD = $\angle$EOC. As shown in part (iv), $\angle$EOC is obtuse ($> 90^\circ$) and not $180^\circ$. So, they do not form a linear pair.

All three pairs are adjacent and do not form a linear pair.

Therefore, adjacent angles that do not form a linear pair are ($\angle$AOB, $\angle$AOE); ($\angle$AOE, $\angle$EOD); ($\angle$EOD, $\angle$COD).

Solution:

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We need to identify the property that relates the angles when two lines are intersected by a transversal.

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(i) If a || b, then $\angle 1 = \angle 5$.

In the given figure, $\angle 1$ and $\angle 5$ are in corresponding positions (upper left corner at each intersection point).

The property used is the Corresponding Angles Property.

This property states that if two parallel lines are intersected by a transversal, then each pair of corresponding angles is equal.

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(ii) If $\angle 4 = \angle 6$, then a || b.

In the given figure, $\angle 4$ and $\angle 6$ are interior angles on opposite sides of the transversal.

The property used is the Alternate Interior Angles Property.

This property (or its converse) states that if a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

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(iii) If $\angle 4 + \angle 5 = 180^\circ$, then a || b.

In the given figure, $\angle 4$ and $\angle 5$ are interior angles on the same side of the transversal.

The property used is the Consecutive Interior Angles Property (or Interior Angles on the Same Side of the Transversal Property).

This property (or its converse) states that if a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary (their sum is $180^\circ$), then the two lines are parallel.

Solution:

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In the given figure, lines 'a' and 'b' are intersected by the transversal line 'c'. Let's identify the different pairs of angles formed.

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(i) The pairs of corresponding angles.

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

The pairs of corresponding angles are:

($\angle 1, \angle 5$); ($\angle 2, \angle 6$); ($\angle 3, \angle 7$); ($\angle 4, \angle 8$)

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(ii) The pairs of alternate interior angles.

Alternate interior angles are interior angles on opposite sides of the transversal.

The pairs of alternate interior angles are:

($\angle 2, \angle 8$); ($\angle 3, \angle 5$)

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(iii) The pairs of interior angles on the same side of the transversal.

These are interior angles on the same side of the transversal.

The pairs of interior angles on the same side of the transversal are:

($\angle 2, \angle 5$); ($\angle 3, \angle 8$)

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(iv) The vertically opposite angles.

Vertically opposite angles are formed by two intersecting lines and are opposite to each other at the point of intersection. They are always equal.

At the intersection of lines 'a' and 'c', the pairs are:

($\angle 1, \angle 3$); ($\angle 2, \angle 4$)

At the intersection of lines 'b' and 'c', the pairs are:

($\angle 5, \angle 7$); ($\angle 6, \angle 8$)

Solution:

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Given that lines p and q are parallel (p || q), and they are intersected by a transversal line.

We are given one angle measuring $125^\circ$ at the intersection with line p. We need to find the measures of the angles labelled a, b, c, d, e, and f.

Since p || q, we can use the properties of angles formed by a transversal intersecting parallel lines.

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By observing the figure and using the property of corresponding angles:

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Now, let's consider the angles at the intersection of line p with the transversal.

Angle $\angle e$ and the $125^\circ$ angle are adjacent angles on the straight line p. They form a linear pair, so their sum is $180^\circ$.

Subtracting $125^\circ$ from both sides:

$\angle e = 55^\circ$

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From the rule of vertically opposite angles at the intersection with line p:

Since $\angle e = 55^\circ$ (from i),

$\angle f = 55^\circ$

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From the rule of vertically opposite angles at the intersection with line q:

Since $\angle d = 125^\circ$,

$\angle b = 125^\circ$

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By the property of corresponding angles, relating angles at line q and line p:

Since $\angle f = 55^\circ$,

$\angle c = 55^\circ$

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By the property of corresponding angles, relating angles at line q and line p:

Since $\angle e = 55^\circ$,

$\angle a = 55^\circ$

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The unknown angles are:

$\angle a = 55^\circ$

$\angle b = 125^\circ$

$\angle c = 55^\circ$

$\angle d = 125^\circ$

$\angle e = 55^\circ$

$\angle f = 55^\circ$

Solution:

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(i) Figure 1:

Given that line l is parallel to line m (l || m), and they are intersected by a transversal.

We are given one angle is $110^\circ$ and the angle $\angle$x. We need to find the value of x.

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Let us assume the angle adjacent to $\angle$x on the line m is $\angle$y, such that $\angle$y is above line m and on the left side of the transversal. This angle $\angle$y is in the same position relative to line m and the transversal as the $110^\circ$ angle is relative to line l and the transversal.

By the property of corresponding angles, since l || m, the corresponding angles are equal.

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We know that a linear pair is a pair of adjacent angles whose non-common arms form a straight line, and their sum is $180^\circ$.

Angles $\angle$x and $\angle$y are adjacent angles on the straight line m. They form a linear pair.

Substitute the value of $\angle y$ into the equation:

Subtract $110^\circ$ from both sides of the equation:

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

$\angle x = 70^\circ$

The value of x is $70^\circ$.

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(ii) Figure 2:

Given that line l is parallel to line m (l || m), and they are intersected by a transversal.

We are given one exterior angle is $100^\circ$ and one interior angle is $\angle$x. We need to find the value of x.

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By the property of corresponding angles, since l || m, the corresponding angles are equal.

Angle $\angle$x is in the interior, below line m, on the right side of the transversal.

The angle $100^\circ$ is in the exterior, above line l, on the right side of the transversal.

These two angles are in corresponding positions.

The value of x is $100^\circ$.

Given:

In the given figure, the arms of two angles, $\angle$ABC and $\angle$DEF, are parallel.

This means AB $\parallel$ DE and BC $\parallel$ EF.

We are also given that $\angle$ABC = $70^\circ$.

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To Find:

(i) $\angle$DGC

(ii) $\angle$DEF

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Solution:

(i) Finding $\angle$DGC:

Consider the parallel lines AB and DE and the transversal BC.

When a transversal intersects two parallel lines, the corresponding angles are equal.

In this case, $\angle$ABC and $\angle$DGC are corresponding angles.

Therefore,

$\angle$DGC = $70^\circ$

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(ii) Finding $\angle$DEF:

Now consider the parallel lines BC and EF and the transversal DE.

When a transversal intersects two parallel lines, the corresponding angles are equal.

In this case, $\angle$DGC and $\angle$DEF are corresponding angles.

From part (i), we found that $\angle$DGC = $70^\circ$.

Therefore,

$\angle$DEF = $70^\circ$

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Answer:

(i) $\angle$DGC = $70^\circ$

(ii) $\angle$DEF = $70^\circ$

We need to determine if line l is parallel to line m in each figure based on the given angle measures.

We will use the properties of angles formed by a transversal intersecting two lines:

• If corresponding angles are equal, the lines are parallel.
• If alternate interior angles are equal, the lines are parallel.
• If consecutive interior angles are supplementary (sum is $180^\circ$), the lines are parallel.

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Solution:

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(i) Figure 1:

Let us consider the two lines, l and m.

The transversal line intersects l and m.

The angles given are $126^\circ$ and $44^\circ$. These are interior angles on the same side of the transversal.

We know that if the sum of interior angles on the same side of the transversal is $180^\circ$, then the lines are parallel.

Let's find the sum of the given interior angles:

$126^\circ + 44^\circ = 170^\circ$

The sum of interior angles on the same side of the transversal is $170^\circ$.

Since the sum is not equal to $180^\circ$, line l is not parallel to line m.

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(ii) Figure 2:

Let us consider the two lines, l and m.

The transversal line intersects l and m.

Let us assume $\angle$x be the vertically opposite angle formed due to the intersection of the straight line l and transversal. This angle is below line l and to the right of the transversal. The given $75^\circ$ angle is above line l and to the right.

So, $\angle$x $= 75^\circ$ (Vertically opposite angles).

We know that if the sum of interior angles on the same side of the transversal is $180^\circ$, then the lines are parallel.

The interior angles on the same side of the transversal are the angle $\angle$x (below l, right) and the interior angle below line m, right. The given $75^\circ$ is above line m, right. The interior angle below line m, right forms a linear pair with the angle above line m, right if that angle was $75^\circ$.

Let's consider the given angles. One is above line l, right ($75^\circ$). The other is above line m, right ($75^\circ$). These are corresponding angles.

If corresponding angles are equal, lines are parallel. In this figure, corresponding angles are equal ($75^\circ = 75^\circ$).

Thus, line l is parallel to line m.

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(iii) Figure 3:

Let us consider the two lines, l and m. The transversal line intersects l and m.

Let us assume $\angle$x be the vertically opposite angle formed due to the intersection of the Straight line l and transversal line n. This angle is above line l and to the right of the transversal.

We know that the sum of interior angles on the same side of the transversal is $180^\circ$ if the lines are parallel.

The provided solution calculates the sum of the given angles ($123^\circ$ and $57^\circ$):

$123^\circ + \angle x = 123^\circ + 57^\circ = 180^\circ$

The sum of these two angles is $180^\circ$.

Therefore, the sum of interior angles on the same side of the transversal is equal to $180^\circ$.

So, line l is parallel to line m.

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(iv) Figure 4:

Let us consider the two lines, l and m.

The transversal line intersects l and m.

The angles given are $98^\circ$ and $72^\circ$. These are interior angles on the same side of the transversal.

We know that if the sum of interior angles on the same side of the transversal is $180^\circ$, then the lines are parallel.

Let's find the sum of the given interior angles:

$98^\circ + 72^\circ = 170^\circ$

The sum of interior angles on the same side of the transversal is $170^\circ$.

Since the sum is not equal to $180^\circ$, line l is not parallel to line m.