Question

in circle y, what is m∠1? 6° 25° 31° 37°

Explanation:

Step1: Recall the vertical - angles and inscribed - angle theorem

Vertical angles are equal and the measure of an inscribed angle is half the measure of its intercepted arc. Also, when two chords intersect inside a circle, the measure of each angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Let the arcs intercepted by the angles be \(a = 37^{\circ}\) and \(b = 25^{\circ}\).
The measure of \(\angle1\) (and its vertical - angle \(\angle2\)) is given by the formula \(m\angle1=\frac{1}{2}(a + b)\).

Step2: Substitute the values of the arcs

Substitute \(a = 37^{\circ}\) and \(b = 25^{\circ}\) into the formula \(m\angle1=\frac{1}{2}(37^{\circ}+25^{\circ})\).
First, calculate the sum inside the parentheses: \(37^{\circ}+25^{\circ}=62^{\circ}\).
Then, find half of the sum: \(m\angle1=\frac{1}{2}\times62^{\circ}=31^{\circ}\).

Answer:

C. \(31^{\circ}\)