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Sum of Opposite Angles of a Cyclic Quadrilateral

In the given ...

Question

In the given figure; ABC, AEQ and CEP are straight lines. Show that $∠ A P E a n d ∠ C Q E$ are supplementary.

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Solution

Given $ABC, AEQ \ and \ CEP$ are straight lines.

In cyclic quadrilateral $ABFP$

$\angle APE + \angle ABE = 180^o$ … … … … … (i)

Similarly in cyclic quadrilateral $BCQE$

$\angle CQE+\angle CBE = 180^o$ … … … … … (ii)

Adding (i) and (ii) we have

$\angle APE + \angle ABE + \angle CQE+\angle CBE = 360$

$\angle APE + \angle CQE + 180 = 360$

$\Rightarrow \angle APE + \angle CQE = 180$

Hence they are supplementary.

$\\$

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