Inscribed Angles - Problem 2

Given two unknown arc measures, it is possible to find their measures using properties of inscribed angles and isosceles triangles. Recall that the measure of an inscribed angle is half of the measure of its intercepted arc.

Also recall that the sum of all arcs on a circle is 360°. So, if one arc is known, subtract its measure from 360° to find the measure of the other arcs of the circle. Since the inscribed triangle is an isosceles triangle, its base angles are congruent. As a result, the intercepted arcs formed by those angles' intersection with the circle must also be congruent, and are the measure of the other arc subtracted from 360°, divided by two. Therefore, by dividing this arc measure by 2, the result is the measure of a base angle of the inscribed isosceles triangle.