Use the image to answer the question.

Question

Quadrilateral upper Q upper U upper A upper D is drawn within a circle such that all vertices touch the edge of the circle.

In the diagram, the following measures are given: m∠A=93°
, mAU=126°
, and mQU=90°
. Find the m∠U
.

(1 point)
m∠U=
º

Answer 1

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of opposite angles equals 180 degrees.

Given the angles:

• \( m\angle A = 93^\circ \)
• \( m\angle U \) (which we need to find)
• \( m\angle Q = 90^\circ \)
• \( m\angle D = 126^\circ \)

We can find \( m\angle U \) using the supplementary property:

1. First, find \( m\angle D + m\angle U = 180^\circ \): \[ 126^\circ + m\angle U = 180^\circ \]

2. Solving for \( m\angle U \): \[ m\angle U = 180^\circ - 126^\circ = 54^\circ \]

Thus, the measure of angle \( U \) is: \[ m\angle U = 54^\circ \]