Find the measure of angle 𝐵.
In this question, we’re asked to find the measure of one of the angles in the triangle 𝐴𝐵𝐶. In order to do this, we will recall some of the properties of circles. Let’s begin by considering the chords 𝐴𝐵 and 𝐴𝐶. These two chords are equidistant from the center of the circle 𝑀 such that line segment 𝑀𝑋 is equal to line segment 𝑀𝑌. We recall that if two chords are equidistant from the center of a circle, their lengths are equal. This means that chord 𝐴𝐵 is equal in length to chord 𝐴𝐶, and triangle 𝐴𝐵𝐶 is therefore isosceles. In any isosceles triangle, two angles are equal, in this case the angles at 𝐵 and 𝐶.
If we let the measure of angle 𝐵 be 𝑥, then we can calculate this using our knowledge of angles in triangles. We know that angles in a triangle sum to 180 degrees. This means that 𝑥 plus 𝑥 plus 35 degrees is equal to 180 degrees. 𝑥 plus 𝑥 is equal to two 𝑥. And subtracting 35 degrees from both sides, we have two 𝑥 is equal to 145 degrees. We can then divide through by two such that 𝑥 is equal to 72.5 degrees. The measure of angle 𝐵 is 72.5 degrees.