Questions: In the diagram below, ⃝P is circumscribed about quadrilateral ABCD. What is the value of x? A. 80° B. 60° C. 120° D. 30°

Transcript text: In the diagram below, $\odot P$ is circumscribed about quadrilateral $A B C D$. What is the value of $x$ ? A. $80^{\circ}$ B. $60^{\circ}$ C. $120^{\circ}$ D. $30^{\circ}$

Solution

Solution Steps

Step 1: Identify the given information

The problem states that $ O $ is the circumcenter of quadrilateral $ ABCD $, and we need to find the value of $ x $.

Step 2: Understand the properties of a cyclic quadrilateral

A quadrilateral is cyclic if and only if the sum of each pair of opposite angles is $ 180^\circ $.

Step 3: Apply the cyclic quadrilateral property

Given that $ ABCD $ is a cyclic quadrilateral, we can use the property that the sum of the opposite angles is $ 180^\circ $. Therefore, $ \angle A + \angle C = 180^\circ $ and $ \angle B + \angle D = 180^\circ $.

Step 4: Use the given angle measures

From the diagram, we can see that $ \angle A = 60^\circ $ and $ \angle C = x $. Using the property of cyclic quadrilaterals: \[ \angle A + \angle C = 180^\circ \] \[ 60^\circ + x = 180^\circ \]

Step 5: Solve for $ x $

Subtract $ 60^\circ $ from both sides of the equation: \[ x = 180^\circ - 60^\circ \] \[ x = 120^\circ \]