Question
apply properties of quadrilaterals inscribed in a circle to determine the ⌒mAD if ⌒mQU = 55 degrees and ⌒mAU = 112 degrees, while the m/_U AD = 100
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GPT-4o mini
To solve for the measure of arc \( mAD \) in a cyclic quadrilateral (a quadrilateral inscribed in a circle), we can use the following properties:
1. The opposite angles of a cyclic quadrilateral are supplementary, meaning the sum of each pair of opposite angles equals \( 180^\circ \).
2. The measure of an angle formed by two chords (or from an external point) is half the sum of the measures of the arcs intercepted by those angles.
Given:
- \( mQU = 55^\circ \)
- \( mAU = 112^\circ \)
- \( m\angle UAD = 100^\circ \)
Using the fact that the measure of an angle at a point is half the measure of the intercepted arcs, we can say:
\[
m\angle UAD = \frac{mAU + mAD}{2}
\]
Now, we know:
\[
100^\circ = \frac{112^\circ + mAD}{2}
\]
To find \( mAD \), we can solve the equation:
1. Multiply both sides by \( 2 \):
\[
200^\circ = 112^\circ + mAD
\]
2. Subtract \( 112^\circ \) from both sides to isolate \( mAD \):
\[
mAD = 200^\circ - 112^\circ = 88^\circ
\]
Thus, the measure of arc \( mAD \) is \( \mathbf{88^\circ} \).
1. The opposite angles of a cyclic quadrilateral are supplementary, meaning the sum of each pair of opposite angles equals \( 180^\circ \).
2. The measure of an angle formed by two chords (or from an external point) is half the sum of the measures of the arcs intercepted by those angles.
Given:
- \( mQU = 55^\circ \)
- \( mAU = 112^\circ \)
- \( m\angle UAD = 100^\circ \)
Using the fact that the measure of an angle at a point is half the measure of the intercepted arcs, we can say:
\[
m\angle UAD = \frac{mAU + mAD}{2}
\]
Now, we know:
\[
100^\circ = \frac{112^\circ + mAD}{2}
\]
To find \( mAD \), we can solve the equation:
1. Multiply both sides by \( 2 \):
\[
200^\circ = 112^\circ + mAD
\]
2. Subtract \( 112^\circ \) from both sides to isolate \( mAD \):
\[
mAD = 200^\circ - 112^\circ = 88^\circ
\]
Thus, the measure of arc \( mAD \) is \( \mathbf{88^\circ} \).
Answer
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