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:

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

We can find $m\mathrm{\angle }U$ using the supplementary property:

1. First, find $m\mathrm{\angle }D+m\mathrm{\angle }U={180}^{\circ }$: $${126}^{\circ }+m\mathrm{\angle }U={180}^{\circ }$$

2. Solving for $m\mathrm{\angle }U$: $$m\mathrm{\angle }U={180}^{\circ }-{126}^{\circ }={54}^{\circ }$$

Thus, the measure of angle $U$ is: $$m\mathrm{\angle }U={54}^{\circ }$$