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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.

Apply properties of quadrilaterals inscribed in a circle to determine the mAD
if mQU=55°
and mAU=112°
, while the m∠UAD=100°
.

(1 point)
mAD=
°

Answer 1

To solve for \( m\angle ADQ \) in the cyclic quadrilateral \( QUAD \), we can use the property that opposite angles in a cyclic quadrilateral sum to \( 180^\circ \).

Given:

• \( m\angle QUAD = m\angle Q = 55^\circ \)
• \( m\angle AU = 112^\circ \)
• \( m\angle UAD = 100^\circ \)

First, we find the measure of the angle \( m\angle DQU \):

Since \( m\angle UAD + m\angle Q = 180^\circ \): \[ m\angle UAD + m\angle DQU = 180^\circ \] Substituting the known values: \[ 100^\circ + m\angle DQU = 180^\circ \] \[ m\angle DQU = 180^\circ - 100^\circ = 80^\circ \]

Next, we need to find \( m\angle ADQ \):

Using the same property for the other pair of opposite angles in quadrilateral \( QUAD \): \[ m\angle ADQ + m\angle QU = 180^\circ \] Substituting the known values: \[ m\angle ADQ + 55^\circ = 180^\circ \] \[ m\angle ADQ = 180^\circ - 55^\circ = 125^\circ \]

Thus, \( m\angle ADQ = 125^\circ \).

Therefore, the final answer is: \[ m\angle ADQ = 125^\circ \]