Page 152 - Start Up Mathematics_7

(e) ∠x = 120° (Corresponding angles)
(f) ∠a = 50° (Alternate interior angles)

∠b = ∠a = 50° (Corresponding angles)

∠f = ∠b = 50° (Alternate interior angles)
∠d = ∠f = 50° (Vertically opposite angles)

∠e = 180° – ∠d = 180° – 50° = 130° (Linear pair)

∠c = ∠e = 130° (Alternate exterior angles)

Transversal and parallel lines

If a transversal cuts two lines such that:
• a pair of corresponding angles are equal, then the lines are parallel.

• a pair of alternate interior angles are equal, then the lines are parallel.
• a pair of interior angles on the same side of the transversal are supplementary, then the lines
are parallel.

Example 30: Give reasons for each of the following:

(a) Is l || m? Why? (b) Find x. Is l || m? Why? (c) If l || m, find x.
t t t

l l l
50° 50° 70°
x x
50°
m m m
130°

Solution: (a) Yes, l || m as the alternate interior angles are equal.
(b) x = 130° (Vertically opposite angles)

x + 50° = 130° + 50° = 180°
 Consecutive interior angles are supplementary ∴ l || m

(c) x = 180° – 70° = 110° (Consecutive interior angles)

Example 31: Give reasons to justify the following statements:
(a) If p || q, then ∠4 = ∠6. 1 2 p
4 3
(b) If ∠3 = ∠7, then p || q.
5 6
(c) If ∠4 + ∠5 = 180°, then p || q. 8 7 q
Solution: (a) Since the lines are parallel, ∴ the alternate interior
angles are equal.

(b) Since the corresponding angles are equal, ∴ the lines are parallel.

(c) Since pair of consecutive interior angles is supplementary, ∴ the lines are parallel.

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