Question

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

Answer 1

To solve for \( m \angle ADU \) in a cyclic quadrilateral where the vertices are inscribed in a circle, we can use the properties of inscribed angles and opposite angles.

In a cyclic quadrilateral, the opposite angles are supplementary. Therefore, we have:

\[
m \angle AUD + m \angle UAD = 180°
\]

We are given:

- \( m \angle UAD = 100° \)

Thus,
\[
m \angle AUD = 180° - m \angle UAD = 180° - 100° = 80°
\]

Next, considering the angles in the quadrilateral \( A, U, D, Q \):

1. \( m \angle AUD = 80° \)
2. \( m \angle QU = 55° \)
3. \( m \angle AUQ = 112° \)

Now, we need to find \( m \angle ADU \). In the cyclic quadrilateral \( ADQU \), the angles \( \angle AUQ \) and \( \angle ADU \) are also supplementary, since they are opposite angles:

\[
m \angle AUQ + m \angle ADU = 180°
\]

Substituting the known value of \( m \angle AUQ \):

\[
112° + m \angle ADU = 180°
\]

Now, solving for \( m \angle ADU \):

\[
m \angle ADU = 180° - 112° = 68°
\]

Thus, the measure of \( m \angle ADU \) is:

\[
\boxed{68°}
\]

Answer 2

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