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Home » 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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