In the given circle, suppose arcRQT = 106 and m

In the given circle, suppose ${\color{Red} m\widehat{RQT}=106^{\circ}}$ and ${\color{Red} m\angle QRS = 73^{\circ}}$ Find the following.

Solution:

We know that,

Measure of inscribed angle = 1/2 × measure of intercepted arc

$\therefore \angle RST = \frac{1}{2}\times \widehat{RQT}$

$\therefore \angle RST = \frac{1}{2}\times 106 = 53^{\circ}$

Now look at the definition of cyclic quadrilateral.

A cyclic quadrilateral is a four sided shape that can be inscribed into a circle. Each vertex of the quadrilateral lies on the circumference of the circle and is connected by four chords.

Therefore given quadrilateral STQR is a cyclic quadrilateral.

And using the property of cyclic quadrilateral ‘The opposite angles of a cyclic quadrilateral have a total of $180°.$ $180° ‘$

we can easily find the remaining two angles.

$\angle RST +\angle RQT =180$

$53 +\angle RQT =180$

$\angle RQT =180-53=127^{\circ}$

Similarly

$\angle QTS =180-73=107^{\circ}$

$\mathbf{m\angle RQT =127^{\circ}}$

$\mathbf{m\angle QTS =107^{\circ}}$

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