7 th maths chapter 5 || 5.2

7th Standard Mathematics — Chapter 5: Geometry

Exercise 5.2 Solutions (Parallel Lines and Transversals)

1. From the figures, name the marked pair of angles

1. Figure (i):
   Answer: Interior alternate angles
   Reason: Angles 1 and 2 lie on opposite sides of the transversal line l inside the lines m and n.
2. Figure (ii):
   Answer: Exterior alternate angles
   Reason: Angles 1 and 2 lie on opposite sides of the transversal line l outside the lines m and n.
3. Figure (iii):
   Answer: Interior angles on the same side of the transversal
   Reason: Both angles lie on the same side of the transversal line l inside lines m and n.
4. Figure (iv):
   Answer: Corresponding angles
   Reason: Angles 1 and 2 lie in matching positions relative to the transversal line l and lines m, n.
5. Figure (v):
   Answer: Vertically opposite angles
   Reason: Angles 1 and 2 are directly opposite each other at the intersection of two lines.
6. Figure (vi):
   Answer: Corresponding angles
   Reason: Angles 1 and 2 occupy identical relative positions at each intersection along the transversal line l.

2. Find the measure of angle x in each of the following figures

1. Figure (i):
   The marked angles (35° and x) are corresponding angles.
   x = 35°
2. Figure (ii):
   The marked angles (65° and x) are corresponding angles.
   x = 65°
3. Figure (iii):
   145° and x are interior alternate angles.
   x = 145°
4. Figure (iv):
   135° and x are exterior alternate angles.
   x = 135°
5. Figure (v):
   The line l is perpendicular to parallel lines m and n, making the corresponding angle a right angle (90°).
   x = 90°

3. Find the measure of angle y in each of the following figures

1. Figure (i):
   28° and y are interior angles on the same side of the transversal, so they are supplementary:
   28° + y = 180°
   y = 180° - 28° = 152°
2. Figure (ii):
   58° and y are exterior alternate angles.
   y = 58°
3. Figure (iii):
   123° and y are interior angles on the same side of the transversal (supplementary):
   123° + y = 180°
   y = 180° - 123° = 57°
4. Figure (iv):
   108° and y are corresponding angles.
   y = 108°

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4. Find the measure of angle \( z \) in each of the following figures.

(i) \( z = 31^\circ \)

Reason: Alternate interior angles between parallel lines \( m \) and \( n \) are equal.

(ii) \( z = 135^\circ \)

Reason: Corresponding angles between parallel lines \( m \) and \( n \) are equal.

(iii) \( z = 101^\circ \)

Reason: The corresponding angle to \( 79^\circ \) on line \( n \) is \( 79^\circ \). Angle \( z \) and \( 79^\circ \) form a linear pair on line \( n \):
\( z + 79^\circ = 180^\circ \implies z = 180^\circ - 79^\circ = 101^\circ \)

(iv) \( z = 158^\circ \)

Reason: Corresponding angle at line \( n \) is \( 22^\circ \). Angle \( z \) forms a linear pair with it:
\( z + 22^\circ = 180^\circ \implies z = 180^\circ - 22^\circ = 158^\circ \)

5. Find the value of angle \( a \) in each of the following figures.

(i) \( a = 18^\circ \)

Calculation: The interior angle corresponding to \( 126^\circ \) on line \( m \) forms a linear pair with \( 3a \):
\( 3a + 126^\circ = 180^\circ \implies 3a = 54^\circ \implies a = \frac{54^\circ}{3} = 18^\circ \)

(ii) \( a = 30.5^\circ \)

Calculation: \( 4a + 13^\circ \) and \( 135^\circ \) are corresponding angles:
\( 4a + 13^\circ = 135^\circ \implies 4a = 122^\circ \implies a = \frac{122^\circ}{4} = 30.5^\circ \)

(iii) \( a = 2^\circ \)

Calculation: \( 8a + 29^\circ \) and \( 45^\circ \) are alternate exterior angles:
\( 8a + 29^\circ = 45^\circ \implies 8a = 16^\circ \implies a = \frac{16^\circ}{8} = 2^\circ \)

(iv) \( a = 15^\circ \)

Calculation: Transversal line is perpendicular to parallel lines:
\( 6a = 90^\circ \implies a = \frac{90^\circ}{6} = 15^\circ \)

6. Find the value of angle \( x \) in both the figures.

(i) \( x = 55^\circ \)

Calculation: Alternate interior angles are equal:
\( 3x - 40^\circ = 2x + 15^\circ \implies 3x - 2x = 15^\circ + 40^\circ \implies x = 55^\circ \)

(ii) \( x = 35^\circ \)

Calculation: Corresponding angle of \( (2x - 15^\circ) \) and \( (3x + 20^\circ) \) form a linear pair:
\( (2x - 15^\circ) + (3x + 20^\circ) = 180^\circ \implies 5x + 5^\circ = 180^\circ \implies 5x = 175^\circ \implies x = 35^\circ \)

7. Anbu's marked angles check:

(i) Correct

Reason: Corresponding angles are equal (\( 105^\circ = 105^\circ \), \( 75^\circ = 75^\circ \)) and interior angles on the same side sum to \( 180^\circ \) (\( 105^\circ + 75^\circ = 180^\circ \)).

(ii) Incorrect

Reason: Vertically opposite angles at line \( n \) are marked as \( 120^\circ \) and \( 60^\circ \), which is impossible because vertically opposite angles must be equal (\( 120^\circ \neq 60^\circ \)).

8. Mention two real-life situations where we use parallel lines:

1. Railway Tracks: Steel rails are kept parallel to maintain a fixed distance so trains can move safely without derailing.
2. Zebra Crossings / Ladder Rungs: The painted stripes on pedestrian crossings and the steps of a ladder are placed parallel for uniform structure and safety.

9. Minimum number of angles needed to find remaining angles:

Answer: 1 Angle

Reason: Knowing just 1 angle allows us to determine all other 7 angles because:

• Its vertically opposite angle is equal.
• Its adjacent angles form a linear pair (subtract from \( 180^\circ \)).
• All 4 angles on the second line can be derived using corresponding or alternate angle properties.

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