Question

quadrilateral bcde is inscribed in circle a as shown. what is m∠e?
a. 84°
b. 96°
c. 89°
d. 111°

Explanation:

Step1: Recall cyclic quadrilateral property

In a cyclic quadrilateral, the sum of opposite angles is \(180^\circ\). Wait, no, actually, for a cyclic quadrilateral (inscribed in a circle), the sum of a pair of opposite angles is \(180^\circ\)? Wait, no, correction: In a cyclic quadrilateral, the sum of each pair of opposite angles is \(180^\circ\)? Wait, no, let's check. Wait, the quadrilateral here is BCDE inscribed in circle A. So angles at B, C, D, E. Wait, angle at B is \(69^\circ\), angle at C is \(84^\circ\). Wait, maybe we need to find angle E. Wait, maybe the quadrilateral has angles at B, C, D, E. Wait, perhaps the sum of all interior angles of a quadrilateral is \(360^\circ\), but since it's cyclic, opposite angles sum to \(180^\circ\)? Wait, no, cyclic quadrilateral: opposite angles are supplementary (sum to \(180^\circ\))? Wait, no, actually, in a cyclic quadrilateral, the measure of an angle is equal to \(180^\circ\) minus the measure of its opposite angle. Wait, let's see the given angles. Angle at B: \(69^\circ\), angle at C: \(84^\circ\). Wait, maybe angle E is opposite to angle C? No, wait, let's label the quadrilateral: B, C, D, E. So sides: BC, CD, DE, EB. So angles: \(\angle B\), \(\angle C\), \(\angle D\), \(\angle E\). Wait, maybe \(\angle B\) and \(\angle D\) are opposite, \(\angle C\) and \(\angle E\) are opposite? Wait, no, that might not be. Wait, maybe the sum of all angles in a quadrilateral is \(360^\circ\), but since it's cyclic, \(\angle B + \angle D = 180^\circ\) and \(\angle C + \angle E = 180^\circ\)? Wait, no, that's not correct. Wait, cyclic quadrilateral: the sum of a pair of opposite angles is \(180^\circ\). So if \(\angle B = 69^\circ\), then its opposite angle \(\angle D = 180 - 69 = 111^\circ\)? No, that's option D. But we need \(\angle E\). Wait, \(\angle C = 84^\circ\), so its opposite angle \(\angle E = 180 - 84 = 96^\circ\)? Wait, that's option B. Let's check: sum of angles in quadrilateral: \(69 + 84 + 111 + 96 = 69+84=153, 111+96=207, 153+207=360\), which is correct. So \(\angle E = 180 - 84 = 96^\circ\)? Wait, no, wait, if \(\angle C\) and \(\angle E\) are opposite, then \(\angle C + \angle E = 180^\circ\), so \(\angle E = 180 - 84 = 96^\circ\). So that's option B.

Step2: Apply cyclic quadrilateral property

In a cyclic quadrilateral, opposite angles are supplementary (sum to \(180^\circ\)). Given \(\angle C = 84^\circ\), and \(\angle E\) is opposite to \(\angle C\), so \(m\angle E = 180^\circ - 84^\circ = 96^\circ\).

Answer:

B. \(96^\circ\)