Question

question 9 of 10 what is the measure of ∠abc? 131° 53° a. 131° b. 127° c. 92° d. 88°

Explanation:

Step1: Recall circle - angle property

The sum of the measures of the arcs of a circle is 360°.

Step2: Find the measure of arc $DE$

Let the measure of arc $DE=x$. Then $131^{\circ}+53^{\circ}+x = 360^{\circ}$. So $x=360^{\circ}-(131^{\circ}+53^{\circ})=360^{\circ}-184^{\circ}=176^{\circ}$.

Step3: Use the angle - arc relationship for angles formed by two chords

The measure of an angle formed by two chords in a circle is half the sum of the measures of the intercepted arcs. $\angle ABC=\frac{1}{2}(\text{arc }AC+\text{arc }DE)$. The measure of arc $AC = 53^{\circ}$ and arc $DE = 176^{\circ}$. So $\angle ABC=\frac{1}{2}(53^{\circ}+176^{\circ})=\frac{1}{2}(229^{\circ}) = 92^{\circ}$ (using the fact that the non - overlapping arcs related to the angle formed by the chords).

Answer:

C. 92°