Question

ii. Prove the theorem: The opposite angles formed by two intersecting lines are of
equal measures.
​

Answer 1

Step-by-step explanation:

Vertical Angles: Theorem and Proof

Theorem: In a pair of intersecting lines the vertically opposite angles are equal.

Proof: Consider two lines AB←→ and CD←→ which intersect each other at O. The two pairs of vertical angles are:

i) ∠AOD and ∠COB

ii) ∠AOC and ∠BOD

It can be seen that ray OA¯¯¯¯¯¯¯¯ stands on the line CD←→ and according to Linear Pair Axiom, if a ray stands on a line, then the adjacent angles form a linear pair of angles.

Therefore, ∠AOD + ∠AOC = 180° —(1) (Linear pair of angles)

Similarly, OC¯¯¯¯¯¯¯¯ stands on the line AB←→.

Therefore, ∠AOC + ∠BOC = 180° —(2) (Linear pair of angles)

From (1) and (2),

∠AOD + ∠AOC = ∠AOC + ∠BOC

⇒ ∠AOD = ∠BOC —(3)

Also, OD¯¯¯¯¯¯¯¯ stands on the line AB←→.

Therefore, ∠AOD + ∠BOD = 180° —(4) (Linear pair of angles)

From (1) and (4),

∠AOD + ∠AOC = ∠AOD + ∠BOD

⇒ ∠AOC = ∠BOD —(5)

Thus, the pair of opposite angles are equal.

Hence, proved.

Solved Example

Consider the figure given below to understand this concept.

Vertically Opposite Angles - Example

In the given figure ∠AOC = ∠BOD and ∠COB = ∠AOD(Vertical Angles)

⇒ ∠BOD = 105° and ∠AOD = 75°