Question

quadrilateral abcd is inscribed in this circle. what is the measure of ∠a? enter your answer in the box.

Explanation:

Step1: Recall cyclic quadrilateral property

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary, meaning their sum is \(180^\circ\). Also, an inscribed angle subtended by a diameter is \(90^\circ\), but here we use the property that \(\angle A\) and \(\angle C\) (the given \(43^\circ\) angle? Wait, no, wait. Wait, actually, in cyclic quadrilateral \(ABCD\), \(\angle A\) and \(\angle C\) are opposite? Wait, no, let's check the diagram. Wait, the angle at \(C\) is \(43^\circ\), but actually, in a cyclic quadrilateral, the sum of a pair of opposite angles is \(180^\circ\). Wait, but also, if we consider that \(\angle D\) and \(\angle B\) are right angles? Wait, no, maybe the angle at \(C\) is \(43^\circ\), and \(\angle A\) is opposite to the angle that is supplementary? Wait, no, let's correct. Wait, in a cyclic quadrilateral, opposite angles are supplementary. So if \(\angle C = 43^\circ\)? No, wait, maybe the angle at \(C\) is \(43^\circ\), but actually, the angle \(\angle A\) and the angle opposite to it (let's say \(\angle C\) is not, wait, maybe the triangle? Wait, no, quadrilateral \(ABCD\) is cyclic. So \(\angle A + \angle C = 180^\circ\)? Wait, no, maybe the angle at \(C\) is \(43^\circ\), but that's not the opposite angle. Wait, maybe I made a mistake. Wait, actually, in a cyclic quadrilateral, the sum of each pair of opposite angles is \(180^\circ\). So if \(\angle C\) is \(43^\circ\), but that's not the case. Wait, maybe the angle given is \(\angle BCD = 43^\circ\), and \(\angle A\) is supplementary to the angle that is related? Wait, no, let's think again. Wait, in a cyclic quadrilateral, \(\angle A + \angle C = 180^\circ\), where \(\angle C\) is the angle at vertex \(C\). Wait, but the diagram shows angle at \(C\) is \(43^\circ\)? No, that can't be. Wait, maybe the angle at \(C\) is \(43^\circ\), but actually, the correct property is that in a cyclic quadrilateral, opposite angles are supplementary. So if \(\angle C = 43^\circ\), then \(\angle A = 180^\circ - 43^\circ = 137^\circ\)? Wait, no, that doesn't seem right. Wait, maybe the angle at \(C\) is \(43^\circ\), but that's an inscribed angle? Wait, no, the quadrilateral is cyclic, so \(\angle A\) and \(\angle C\) are opposite angles, so they should be supplementary. Wait, let's confirm: In a cyclic quadrilateral, the sum of a pair of opposite angles is \(180^\circ\). So if \(\angle C = 43^\circ\), then \(\angle A = 180 - 43 = 137\)? Wait, but maybe the angle at \(C\) is not \(43^\circ\), but the angle between \(CD\) and \(CB\) is \(43^\circ\), and \(\angle A\) is supplementary to that? Wait, no, let's check the diagram again. The diagram shows quadrilateral \(ABCD\) inscribed in a circle, with angle at \(C\) being \(43^\circ\). So \(\angle A\) and \(\angle C\) are opposite angles, so they sum to \(180^\circ\). Therefore, \(\angle A = 180^\circ - 43^\circ = 137^\circ\).

Step2: Calculate \(\angle A\)

Using the property of cyclic quadrilaterals (opposite angles are supplementary):
\[
\angle A + \angle C = 180^\circ
\]
Given \(\angle C = 43^\circ\), substitute:
\[
\angle A = 180^\circ - 43^\circ = 137^\circ
\]

Answer:

\(137\)