Question

PQ is tangent to the circle at C. In the circle, m (The figure is not drawn to scale.)

Answer 1

m∠DCQ=56°

Explanation:

step 1

Find the measure of arc ABC

we know that

The inscribed angle is half that of the arc it comprises

so

m∠D=(1/2)[arc ABC]

we have

m∠D=78°

substitute and solve for arc ABC

78°=(1/2)[arc ABC]

156°=[arc ABC]

Rewrite

arc ABC=156°

step 2

Find the measure of arc DC

we know that

arc ABC+arc AD+arc DC=360° -----< by complete circle

substitute the given values

156°+92°+arc DC=360°

arc DC=360°-248°

arc DC=112°

step 3

Find the measure of angle DCQ

we know that

The inscribed angle is half that of the arc it comprises

so

m∠DCQ=(1/2)[arc DC]

we have

arc DC=112°

substitute

m∠DCQ=(1/2)[112°]=56°