Question

angles in the same segment of a circle are equal. prove this theorem

Answer 1

Given - A circle with centre at O .

To Prove - angle poq = 2 angle paq .

Proof - angle poq = angle paq -------- 1.

Similarly

POQ =2 PBQ

From 1

2 PBQ = 2 PAQ

Angle PBQ = Angle PAQ .

HENCE PROVED...

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Answer 2

Answer:

Hence, proved.

Step-by-step explanation:

Given: Angles in the same segment of a circle are equal.

We have to prove the above theorem.

We are solving in the following way:

We have,

Angles in the same segment of a circle are equal.

Let assume,

A circle with center at  O. Points  P & Q on this circle subtends angles angle PAQ  and angle PBQ  at points A and B respectively.

Chord PQ subtends angle POQ at the center in the bellow figure.

From Theorem: Angle subtended by an arc at the center is double the angle subtended by it at any other point on the circle.

$\therefore \angle \mathrm{POQ}=2 \angle \mathrm{PAQ}\ \ ..1)$

$\angle \mathrm{POQ}=2 \angle \mathrm{PBQ} \quad \ldots(2)$

From 1) and 2)

$\begin{array}{l}2 \angle \mathrm{PBQ}=2 \angle \mathrm{PAQ} \\\angle \mathrm{PBQ}=\angle \mathrm{PAQ}\end{array}$

Hence proved.