Question

Prove that if two lines intersect each other, then the vertically opposite angles are equal.​

Answer 1

Answer:

We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.

Now, ray OA stands on line CD.

Therefore, ∠ AOC + ∠ AOD = 180° (Linear pair axiom) ………..(1)

Can we write ∠ AOD + ∠ BOD = 180°? (Linear pair axiom)……………(2)

From (1) and (2), we can write

∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD

This implies that ∠ AOC = ∠ BOD

Similarly, it can be proved that ∠AOD = ∠BOC

Answer 2

Answer:

Step-by-step explanation:

Construction: Draw OL and OM perpendiculars to chords AB and CD respectively. Join OP.

To prove ∠OPL=∠OPM

Proof: Consider triangles OLP and OMP, OL=OM[ Equal chords AB and DD are equidistant from the centre of the circle]

OP is common.

∠OLP=∠OMP[90 each]∠OLP=∠OMP

ΔOLP≅ΔOMP [RHS]

∠OPL=∠OPM.[CPCT]