GSEB Class 7 Maths Solutions Chapter 5 Lines and Angles Exercise 5.2

Question 1. State the property that is used in each of the following statements?
(i) If a \( \parallel \) b, then \( \angle 1 = \angle 5 \).
(ii) If \( \angle 4 = \angle 6 \), then a \( \parallel \) b.
(iii) If \( \angle 4 + \angle 5 = 180^{\circ} \), then a \( \parallel \) b.
Answer:
(i) If two straight lines are made parallel and then crossed by another line (called a transversal), the angles that are in the same relative position (corresponding angles) will always be equal.
(ii) If two given lines are crossed by a transversal line in such a way that the angles inside and on opposite sides of the transversal (interior alternate angles) are equal, then the two lines must be parallel.
(iii) If two given lines are cut by a transversal so that the total sum of the inside angles on the same side of the transversal is \( 180^{\circ} \), then those lines are parallel.

Exam Tip: Remember the specific names for angle pairs formed by parallel lines and a transversal (corresponding, alternate interior, consecutive interior) and their associated properties.

Question 2. In the adjoining figure, identify:
(i) the pairs of corresponding angles.
(ii) the pairs of alternate interior angles.
(iii) the pairs of interior angles on the same side of the transversal.
(iv) the vertically opposite angles.
Answer:
(i) The pairs of corresponding angles are: \( (\angle 1, \angle 5); (\angle 2, \angle 6); (\angle 3, \angle 7) \) and \( (\angle 4, \angle 8) \). These angles are found in similar positions at each intersection.
(ii) The pairs of alternate interior angles are: \( (\angle 2, \angle 8) \) and \( (\angle 3, \angle 5) \). These angles are between the parallel lines but on opposite sides of the transversal.
(iii) The pairs of interior angles on the same side of the transversal are: \( (\angle 2, \angle 5) \) and \( (\angle 3, \angle 8) \). These angles are inside the parallel lines and on the same side of the cutting line.
(iv) The vertically opposite angles are: \( (\angle 1, \angle 3); (\angle 2, \angle 4); (\angle 5, \angle 7) \) and \( (\angle 6, \angle 8) \). These pairs of angles are opposite each other when two lines cross.

Exam Tip: Clearly label the angles in your diagram and use the definitions to correctly identify each pair of angles.

Question 3. In the adjoining figures p \( \parallel \) q. Find the unknown angles.
Answer:
We are given that lines p and q are parallel. We also see that \( \angle e + 125^{\circ} = 180^{\circ} \) because they form a linear pair.
\( \implies \angle e = 180^{\circ} - 125^{\circ} \)
\( \implies \angle e = 55^{\circ} \)
Since \( \angle e \) and \( \angle f \) are vertically opposite angles, they are equal.
\( \implies \angle f = 55^{\circ} \)
Also, \( \angle a \) and \( \angle e \) are corresponding angles, so they are equal.
\( \implies \angle a = 55^{\circ} \)
The angle \( \angle b \) and the \( 125^{\circ} \) angle are alternate exterior angles, so \( \angle b = 125^{\circ} \).
Since \( \angle b \) and \( \angle c \) form a linear pair, their sum is \( 180^{\circ} \).
\( \implies \angle b + \angle c = 180^{\circ} \)
\( \implies 125^{\circ} + \angle c = 180^{\circ} \)
\( \implies \angle c = 180^{\circ} - 125^{\circ} \)
\( \implies \angle c = 55^{\circ} \)
Finally, \( \angle b \) and \( \angle d \) are vertically opposite angles, so they are equal.
\( \implies \angle d = \angle b = 125^{\circ} \)
Therefore, the required angle measurements are: \( \angle a = 55^{\circ}, \angle b = 125^{\circ}, \angle c = 55^{\circ}, \angle d = 125^{\circ}, \angle e = 55^{\circ}, \angle f = 55^{\circ} \).

Exam Tip: Systematically find each angle using the properties of linear pairs, vertically opposite angles, corresponding angles, and alternate exterior angles when lines are parallel.

Question 4. Find the value of x in each of the following figures if l \( \parallel \) m.
(i)
(ii)
Answer:
(i) In this figure, line l is parallel to line m, and t is the transversal. Let the angle vertically opposite to \( 110^{\circ} \) be \( \angle p \).
\( \implies \angle p = 110^{\circ} \) (Vertically opposite angles)
The angles \( \angle p \) and \( x \) are consecutive interior angles. When two parallel lines are cut by a transversal, the sum of consecutive interior angles is \( 180^{\circ} \).
\( \implies \angle p + x = 180^{\circ} \)
\( \implies 110^{\circ} + x = 180^{\circ} \)
\( \implies x = 180^{\circ} - 110^{\circ} \)
\( \implies x = 70^{\circ} \)
Thus, the required value of x is \( 70^{\circ} \).
(ii) In this figure, lines l and m are parallel, and line a is the transversal. The angle \( x \) and the \( 100^{\circ} \) angle are corresponding angles. Because lines l and m are parallel, corresponding angles are equal.
\( \implies x = 100^{\circ} \)
Thus, the required value of x is \( 100^{\circ} \).

Exam Tip: When lines are parallel, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (add up to 180°).

Question 5. In the given figure, the arms of two angles are parallel. If \( \angle ABC = 70^{\circ} \), then find:
(i) \( \angle DGC \)
(ii) \( \angle DEF \)
Answer:
We are given that \( AB \parallel ED \) and \( BC \parallel EF \).
(i) When \( AB \parallel ED \) and \( BC \) acts as a transversal line:
The angles \( \angle DGC \) and \( \angle ABC \) are corresponding angles. Since the lines are parallel, these angles are equal.
\( \implies \angle DGC = \angle ABC \)
Given that \( \angle ABC = 70^{\circ} \).
\( \implies \angle DGC = 70^{\circ} \)
(ii) When \( BC \parallel EF \) and \( ED \) acts as a transversal line:
The angles \( \angle DEF \) and \( \angle DGC \) are corresponding angles. Since the lines are parallel, these angles are equal.
\( \implies \angle DEF = \angle DGC \)
From part (i), we found that \( \angle DGC = 70^{\circ} \).
\( \implies \angle DEF = 70^{\circ} \)

Exam Tip: When dealing with parallel lines and transversals, always carefully identify which lines are parallel and which line is acting as the transversal for each pair of angles.

Question 6. In the given figures below, decide whether l is parallel to m.
Answer:
(i) The sum of the interior angles on the same side of the transversal is \( 44^{\circ} + 126^{\circ} = 170^{\circ} \).
Since \( 170^{\circ} \neq 180^{\circ} \), the sum of these angles is not \( 180^{\circ} \). Therefore, lines l and m are not parallel.
(ii) Let the angle vertically opposite to the given \( 75^{\circ} \) angle be \( \angle p \). So, \( \angle p = 75^{\circ} \) (Vertically opposite angles). The interior angles on the same side of the transversal are \( \angle p \) and the other \( 75^{\circ} \) angle. Their sum is \( 75^{\circ} + 75^{\circ} = 150^{\circ} \).
Since \( 150^{\circ} \neq 180^{\circ} \), the lines l and m are not parallel.
(iii) Let the angle forming a linear pair with \( 123^{\circ} \) be \( \angle p \).
\( \implies \angle p + 123^{\circ} = 180^{\circ} \) (Linear pair)
\( \implies \angle p = 180^{\circ} - 123^{\circ} \)
\( \implies \angle p = 57^{\circ} \)
The angle \( \angle p \) and the angle marked \( 57^{\circ} \) (which is corresponding to \( \angle p \)) are equal. Since corresponding angles are equal, lines l and m are parallel.
(iv) Let the angle forming a linear pair with \( 98^{\circ} \) be \( \angle 1 \).
\( \implies \angle 1 + 98^{\circ} = 180^{\circ} \) (Linear pair)
\( \implies \angle 1 = 180^{\circ} - 98^{\circ} \)
\( \implies \angle 1 = 82^{\circ} \)
The corresponding angle to \( \angle 1 \) is \( \angle 3 = 72^{\circ} \).
Since \( \angle 1 \neq \angle 3 \) (i.e., \( 82^{\circ} \neq 72^{\circ} \)), the corresponding angles are not equal. Therefore, lines l and m are not parallel.

Exam Tip: To determine if lines are parallel, always check if any of the angle pair properties (corresponding, alternate interior, consecutive interior, vertically opposite, linear pair) hold true.