Question

apply properties of quadrilaterals inscribed in a circle to determine the mAD if mQU = 55 degrees and mAU = 112 degrees, while the m angleU AD = 100 degrees. the angle is not 125 degrees

Answers

Answered by GPT-4o mini
To determine \( m\angle AD \) in a cyclic quadrilateral, we can use the property that opposite angles sum to 180 degrees.

Given the angles:
- \( m\angle QU = 55^\circ \)
- \( m\angle AU = 112^\circ \)
- \( m\angle UAD = 100^\circ \)

We need to find \( m\angle AD \). Since \( A, U, Q, D \) are the vertices of a cyclic quadrilateral, the property tells us:

\[
m\angle UAD + m\angle QU = 180^\circ
\]

Now, if we denote \( m\angle AD \) as \( x \), we can also apply the cyclic property:

\[
m\angle AU + m\angle AD = 180^\circ
\]

Now, let's substitute the known values into both equations. Using the angle \( A, U, Q, D \):

1. \( m\angle UAD + m\angle QU = 100^\circ + 55^\circ = 155^\circ \)
2. We also need to find \( x \) using \( m\angle AU + m\angle AD = 180^\circ \):

\[
112^\circ + x = 180^\circ
\]

Now, solving for \( x \):

\[
x = 180^\circ - 112^\circ = 68^\circ
\]

Thus, \( m\angle AD = 68^\circ \).

To sum up:
- \( m\angle AD = 68^\circ \)
- And we can confirm that \( m\angle UAD + m\angle QU \) is correctly calculated not to exceed 180 degrees, meaning all properties meet the quadrilateral criteria.

Therefore, the final answer is:

\[
\boxed{68^\circ}
\]

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