Page 155 - Maths Skills - 7

Lines and Angles 153

The vertically opposite angles are always equal to each other.

In Fig. two lines AB and CD intersect each other at a point O.
As a result of this intersection, four angles ∠1, ∠2, ∠3 and ∠4 are formed. ∠1 and ∠3 form a pair of vertically
opposite angles. ∠2 and ∠4 form another pair of vertically opposite angles. Therefore, ∠1 = ∠3 and ∠2 = ∠4.

Absorbing Facts

Angles at a Point D C
When different rays start from a common vertex or initial point, they form different 3 2 B
angles. These angles are called angles at a point. 4 1
In Fig., we find that, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 360°. 5 A
Thus, the sum of all angles at a point is always 360°. O
E

Let’s Attempt

Example 1: Find the complement of the following angles.

(i) 47° (ii) 55°
Solution: We know that the sum of the measure of an angle and its complement is 90°.

(i) Complement of 47° = 90° – 47° = 43°. (ii) Complement of 55° = 90° – 55° = 35°.

Example 2: Find the supplement of the following angles.
(i) 76° (ii) 115°

Solution: We know that the sum of the measure of an angle and its supplement is 180°.
(i) Supplement of 76° = 180° – 76° = 104°. (ii) Supplement of 115° = 180° – 115° = 65°.

Example 3: Find the value of x in Fig. 1 and 2.

Q
(i) B (ii)
4x
O x
4x 2x R 2x 3x P

C O A
Fig. 1 S
Fig. 2

Solution: (i) ∠AOB + ∠BOC = 180° (ii) ∠POQ + ∠QOR + ∠ROS + ∠SOP = 360°

[Linear pair] [Angles at a point]

⇒ 2x + 4x = 180° ⇒ x + 4x + 2x + 3x = 360°

⇒ 6x = 180° ⇒ 10x = 360°

⇒ x = 30° ⇒ x = 36°

Hence, the value of x is 30°. Hence, the value of x is 36°.