Question

in circle z, what is m∠2? 70° 133° 140° 147°

Explanation:

Step1: Recall the property of inscribed - angles and arcs.

The measure of an inscribed - angle is half the measure of its intercepted arc. Also, the sum of the measures of the arcs of a circle is 360°. First, find the measure of arc $\overset{\frown}{BC}$. Let the measure of arc $\overset{\frown}{BC}=x$. Then $133^{\circ}+x + 147^{\circ}=360^{\circ}$.

Step2: Solve for the measure of arc $\overset{\frown}{BC}$.

$x=360^{\circ}-(133^{\circ}+147^{\circ})=360^{\circ}-280^{\circ}=80^{\circ}$.

Step3: Use the inscribed - angle theorem.

$\angle1$ is an inscribed angle that intercepts arc $\overset{\frown}{AD}$, and $\angle2$ is an inscribed angle that intercepts arc $\overset{\frown}{BC}$. The measure of $\angle1=\frac{1}{2}\text{ measure of arc }\overset{\frown}{AD}$, and the measure of $\angle2=\frac{1}{2}\text{ measure of arc }\overset{\frown}{BC}$. Since the measure of arc $\overset{\frown}{BC} = 140^{\circ}$, then $m\angle2=\frac{1}{2}\times140^{\circ}=70^{\circ}$.

Answer:

$70^{\circ}$