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# In the figure, $\triangle PQR$ is an isosceles triangle with $PQ = PR$ and $m \angle PQR = 35^o$. Find $m \angle QSR$ and $m \angle QTR$."

Given:

In the figure, $\triangle PQR$ is an isosceles triangle with $PQ = PR$ and $m \angle PQR = 35^o$.

To do:

We have to find $m \angle QSR$ and $m \angle QTR$.

Solution:

$\angle PQR = 35^o$

This implies,

$\angle PRQ = 35^o$

$\angle PQR + \angle PRQ + \angle QPR = 180^o$           (Sum of angles of a triangle)

$35^o + 35^o + \angle QPR = 180^o$

$70^o + \angle QPR = 180^o$

$\angle QPR = 180^o - 70^o = 110^o$

$\angle QSR = \angle QPR$             (Angle in the same segment of circles)

$\angle QSR = 110^o$

$PQTR$ is a cyclic quadrilateral.

Therefore,

$\angle QTR + \angle QPR = 180^o$

$\angle QTR + 110^o = 180^o$

$\angle QTR = 180^o -110^o = 70^o$

Hence $\angle QTR = 70^o$.

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Updated on: 10-Oct-2022

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