#### Question

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

#### Solution

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

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 +∠ABE + ∠CBE = 360°

But, ∠ABE + ∠CBE = 180° [Linear pair]

∠APE + ∠CQE + 180° = 360°

⇒ ∠APE + ∠CQE = 360° -180° = 180°

Hence ∠APE and ∠CQE are supplementary.

Is there an error in this question or solution?

Solution In the Given Figure; Abc, Aeq and Cep Are Straight Lines. Show that ∠Ape and ∠Cqe Are Supplementary. Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.