Question

In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.

Answer 1

Given – In the figure, ABC, AEQ and CEP are straight lines

To prove – ∠APE + ∠CQE = 180°

Construction – Join EB

Proof – In cyclic quad ABEP,

∠APE + ∠ABE = 180°    ...(1)

Similarly, in cyclic quad BCQE

∠CQE + ∠CBE = 180°   ...(2)

Adding (1) and (2),

∠APE + ∠ABE + ∠CQE + ∠CBE = 180° + 180° = 360°

`=>` ∠APE + ∠CQE + ∠ABE + ∠CBE = 360°

But, ∠ABE + ∠CBE = 180°   ...[Linear pair]

∴ ∠APE + ∠CQE + 180° = 360°

`=>` ∠APE + ∠CQE = 360° – 180° = 180°

Hence ∠APE and ∠CQE are supplementary.