Question

measure of the angle labeled?

Answer 1

Given:

The figure of a circle.

The measure of arc SR = 75 degrees.

Measure of angle RSQ = 95 degrees.

To find:

The measure of SQ.

Solution:

Let us consider, O be the center of the circle.

According to the central angle theorem, the measure central angle on an arc is twice of inscribed angle on the corresponding angle.

Using the central angle theorem, we get

$m\angle QOR=2\times m\angle RSQ$

$m\angle QOR=2\times (95^\circ)$

$m\angle QOR=190^\circ$

It is the central angle on major arc QR.

We know that the measure of complete central angle is 360 degrees.

The central angle on minor arc QR is:

$360^\circ -190^\circ =170^\circ$

It means the measure of arc RSQ is 170 degrees.

$m(arc RS)+m(arc SQ)=m(arcRSQ)$

$75^\circ+m(arc SQ)=170^\circ$

$m(arc SQ)=170^\circ -75^\circ$

$m(arc SQ)=95^\circ$

Therefore, the measure of arc SQ is 95 degrees.