Mathematics
In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.
Circles
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Answer
Join EB.
We know that,
In cyclic quadrilateral sum of opposite angles = 180°.
In cyclic quad. ABEP
⇒ ∠APE + ∠ABE = 180° …..(1) [Opposite angles of a cyclic quad. are supplementary]
Similarly, in cyclic quad. BCQE
⇒ ∠CQE + ∠CBE = 180° …..(2)
Adding (1) and (2), we have
⇒ ∠APE + ∠ABE + ∠CQE + ∠CBE = 180° + 180°
⇒ ∠APE + ∠ABE + ∠CQE + ∠CBE = 360° …..(3)
From figure,
∠ABE + ∠CBE = 180° [Linear pair]
Putting this value of ∠ABE + ∠CBE in Eq 3 we get,
∠APE + ∠CQE + 180° = 360°
⇒ ∠APE + ∠CQE = 360° - 180°
⇒ ∠APE + ∠CQE = 180°
Hence, proved that ∠APE and ∠CQE are supplementary.
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