We are given a cyclic quadrilateral ABCD inscribed in a circle. We are given that the measure of angle A, $m\angle A = 93^\circ$. We need to find the measure of angle C, $m\angle C$.

1. Problem Description

We are given a cyclic quadrilateral ABCD inscribed in a circle. We are given that the measure of angle A, $m\angle A = 93^\circ$. We need to find the measure of angle C, $m\angle C$.

2. Solution Steps

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.

A property of cyclic quadrilaterals is that opposite angles are supplementary, meaning their measures add up to $180^\circ$.

In cyclic quadrilateral ABCD, angles A and C are opposite angles, therefore they are supplementary.

$$m\angle A + m\angle C = 180^\circ$$

We are given $m\angle A = 93^\circ$. Substituting this into the equation, we get:

$$93^\circ + m\angle C = 180^\circ$$

$$m\angle C = 180^\circ - 93^\circ$$

$$m\angle C = 87^\circ$$

3. Final Answer

87