Question

in the diagram below, ⊙p is circumscribed about quadrilateral abcd. what is the value of x? a. 130° b. 65° c. 50° d. 80°

Explanation:

Step1: Recall cyclic quadrilateral property

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of opposite angles is \(180^\circ\). So, \(\angle A + \angle C = 180^\circ\).

Step2: Substitute known angle

We know \(\angle C = 130^\circ\) and \(\angle A = x^\circ\). So, \(x + 130 = 180\).

Step3: Solve for x

Subtract 130 from both sides: \(x = 180 - 130 = 50\). Wait, no, wait. Wait, actually, in a cyclic quadrilateral, the inscribed angle subtended by the same arc... Wait, no, maybe I made a mistake. Wait, no, the property is that opposite angles in a cyclic quadrilateral are supplementary. Wait, but in the diagram, is angle A and angle C opposite? Wait, quadrilateral ABCD, so vertices in order A, B, C, D. So opposite angles are A and C, B and D. So \(\angle A + \angle C = 180^\circ\). Wait, but if angle C is 130, then angle A is 50? But wait, no, maybe the diagram is different. Wait, no, maybe I confused the property. Wait, no, the correct property is that in a cyclic quadrilateral, the sum of a pair of opposite angles is \(180^\circ\). So if \(\angle C = 130^\circ\), then \(\angle A = 180 - 130 = 50^\circ\)? But wait, the options have 50 as option C. Wait, but let me check again. Wait, maybe the angle at C is 130, so angle at A is 50? Wait, but let's re - examine. Wait, maybe the quadrilateral is cyclic, so \(\angle A + \angle C=180^{\circ}\). So \(x + 130 = 180\), so \(x = 50\). But wait, the options: A is 130, B is 65, C is 50, D is 80. So the answer should be C? Wait, no, wait, maybe I messed up the opposite angles. Wait, maybe angle at C and angle at A are not opposite? Wait, quadrilateral ABCD, so the order is A - B - C - D - A. So the opposite angles are \(\angle A\) and \(\angle C\), \(\angle B\) and \(\angle D\). So yes, they should be supplementary. So \(x + 130 = 180\), so \(x = 50\).

Answer:

C. \(50^{\circ}\)