**Vertically Opposite Angles**

Two angles are called a pair of vertically opposite angles, if their arms form two pairs of opposite rays.

Let two lines AB and CD intersect at a point O. Then, two pairs of** vertically opposite angles** are formed.

(i) ∠AOC and ∠BOD (ii) ∠AOD and ∠BOC

**Theorem 1:**

If two lines intersect then the vertically opposite angles are equal.

**Given:** Two lines AB and CD intersect at a point O.

**To prove:** (i) ∠AOC = ∠BOD, (ii) ∠AOD = ∠BOC

**Proof:** Since ray OA stands on line CD, we have:

∠AOC + ∠AOD = 180° [linear pair].

Again, ray OD stands on line AB.

∴ ∠AOD + ∠BOD = 180° [linear pair]

∴ ∠AOC + ∠AOD = ∠AOD + ∠BOD [each equal to 180°]

∴ ∠AOC = ∠BOD

Similarly, ∠AOD = ∠BOC

**Vertically Opposite Angles Example Problems With Solutions**

**Example 1: **Two lines AB and CD intersect at O. If ∠AOC = 50°, find ∠AOD, ∠BOD and ∠BOC.

**Solution: **∠AOD + ∠AOC = 180° (linear pair)

∠AOD + 50° = 180°

∠AOD = 130°

Also ∠BOD = ∠AOC

(vertically opposite angles)

& ∠BOC = ∠AOD = 130°

(vertically opposite angles)

∵ 130°, 50°, 130°.

**Example 2: **Two lines AB and CD intersect at a point O such that ∠BOC + ∠AOD = 280°, as shown in the figure. Find all the four angles.

**Solution: **∠AOC = ∠BOD = x (Let)

(vertically opposite angles)

∵ ∠AOC + (∠AOD + ∠BOC) + ∠BOD = 360°

⇒ x + 280° + x = 360°

⇒ 2x = 80°

⇒ x = 40°

∵ ∠AOC = ∠BOD = x° = 40°.

and ∠BOC = ∠AOD = 280°/2 = 140°.

**Example 3: **In Fig., lines l_{1} and l_{2} intesect at O, forming angles as shown in the figure. If a = 35º, find the values of b, c, and d.

**Solution: **Since lines l_{1} and l_{2} intersect at O.

Therefore,

∠a = ∠c [Vertically opposite angles]

⇒ ∠c = 35º [∵ ∠a = 35º]

Clearly, ∠a + ∠b = 180º

[Since ∠a and ∠b are angles of a linear pair]

⇒ 35º + ∠b = 180º

⇒ ∠b = 180º – 35º

⇒ ∠b = 145º

Since ∠b and ∠d are vertically opposite angles. Therefore,

∠d = ∠b ⇒ ∠d = 145º [∵ ∠b = 145º]

**Example 4: **In Fig., determine the the value of y.

**Solution: **Since ∠COD and ∠EOF are vertically opposite angles. Therefore,

∠COD = ∠EOF ⇒ ∠COD = 5yº

[∵ ∠EOF = 5yº (Given)]

Now, OA and OB are opposite rays.

∵ ∠AOD + ∠DOC + ∠COB = 180º

⇒ 2yº + 5yº + 5yº = 180º

⇒ 12yº = 180º

⇒ yº = 180º/12 = 15.

Thus, yº = 15.

**Example 5: **In Fig., AB and CD are straight lines and OP and OQ are respectively the bisectors of angles BOD and AOC. Show that the rays OP and OQ are in the same line.

**Solution: **In order to prove that OP and OQ are in the same line, it is sufficient to prove that

∠POQ = 180º.

Now, OP is the bisector of ∠AOC

⇒ ∠1 = ∠6 …(i)

and, OQ is the bisector of ∠AOC

⇒ ∠3 = ∠4 ….(ii)

Clearly, ∠2 and ∠5 are vertically opposite angles.

∵ ∠2 = ∠5 ….(iii)

We know that the sum of the angles formed at a point is 360º.

Therefore,

∠1 +∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360º

⇒ (∠1 + ∠6) + (∠3 + ∠4) + (∠2 + ∠5) = 360º

⇒ 2∠1 + 2∠3 + 2∠2 = 360º

[Using (i), (ii) and (iii)]

⇒ 2(∠1 + ∠3 + ∠2) = 360º

⇒ ∠1 + ∠2 + ∠3 = 180º ⇒ ∠POQ = 180º

Hence, OP and OQ are in the same straight line.

**Example 6: **In Fig., two staright lines PQ and RS intersect each other at O. If ∠POT = 75º, find the values of a, b and c.

**Solution: **Since OR and OS are in the same line. Therefore,

∠ROP + ∠POT + ∠TOS = 180º

⇒ 4bº + 75º + bº = 180º ⇒ 5bº + 75º = 180º

⇒ 5bº = 105º ⇒ bº = 21

Since PQ and RS intersect at O. Therefore,

∠QOS = ∠POR

[Vertically oppsostie angles]

⇒ a = 4b

⇒ a = 4 × 21 = 84 [∵ b = 21]

Now, OR and OS are in the same line. Therefore.

∠ROQ + ∠QOS = 180º [Linear pair]

⇒ 2c + a = 180

⇒ 2c + 84 = 180 [∵ b = 84]

⇒ 2c = 96

⇒ c = 48

Hence, a = 84, b = 21 and c = 48