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In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.
In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.
Circles | Class 10 | Exercise 17C | ICSE | Maths | Selina
In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.

Selina Solutions Concise Class 10 Maths Chapter 17 ex. 17(C) - 15

Solution:

Selina Solutions Concise Class 10 Maths Chapter 17 ex. 17(C) - 16

Join \[EB\]

Then, in cyclic quad. \[ABEP\]

\[\angle APE\text{ }+\angle ABE\text{ }=\text{ }{{180}^{o}}~\ldots ..\text{ }\left( i \right)\][Opposite angles of a cyclic quad. are supplementary]

Similarly, in cyclic quad.BCQE

\[\angle CQE\text{ }+\angle CBE\text{ }=\text{ }{{180}^{o}}~\ldots ..\text{ }\left( ii \right)\][Opposite angles of a cyclic quad. are supplementary]

Adding (i) and (ii), we have

\[\angle APE\text{ }+\angle ABE\text{ }+\angle CQE\text{ }+\angle CBE\]

\[~=\text{ }{{180}^{o~}}+\text{ }{{180}^{o}}~=\text{ }{{360}^{o}}\]

Or,

\[\angle APE\text{ }+\angle ABE\text{ }+\angle CQE\text{ }+\angle CBE\text{ }=\text{ }{{360}^{o}}\]

But, \[\angle ABE\text{ }+\angle CBE\text{ }=\text{ }{{180}^{o}}~\][Linear pair]

\[\angle APE\text{ }+\angle CQE\text{ }+\text{ }{{180}^{o}}~=\text{ }{{360}^{o}}\]

\[\angle APE\text{ }+\angle CQE\text{ }=\text{ }{{180}^{o}}\]

Therefore, \[\angle APE\text{ }and\angle CQE\]are supplementary.

i

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