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

88 isnt the right answer

Answers

Answered by GPT-4o mini
To determine the measure of arc \( \overset{\frown}{AD} \) in a cyclic quadrilateral, we can use the properties of cyclic quadrilaterals. In a cyclic quadrilateral, the opposite angles add up to 180 degrees.

Given:
- \( \overset{\frown}{QU} = 55^\circ \)
- \( \overset{\frown}{AU} = 112^\circ \)
- The measure of angle \( \angle UAD = 100^\circ \)

From the properties of cyclic quadrilaterals, we know:
\[
\angle UAD + \angle UQD = 180^\circ
\]

First, we need to find \( \angle UQD \):
\[
\angle UQD = 180^\circ - \angle UAD = 180^\circ - 100^\circ = 80^\circ
\]

Now, since angles subtended by the same arc are equal, we know that:
\[
\angle UQD = \frac{1}{2} \overset{\frown}{QU} \quad \text{and} \quad \angle UAD = \frac{1}{2} \overset{\frown}{AU}
\]
But here, we need to determine \( \overset{\frown}{AD} \).

To find \( \overset{\frown}{AD} \), we can find the measure of arcs \( \overset{\frown}{QU} \) and \( \overset{\frown}{AU} \).

The Cyclic Quadrilateral Converse states that:
\[
\overset{\frown}{AD} = 360^\circ - (\overset{\frown}{QU} + \overset{\frown}{AU})
\]
Now substituting the values:
\[
\overset{\frown}{AD} = 360^\circ - (55^\circ + 112^\circ)
\]
\[
= 360^\circ - 167^\circ
\]
\[
= 193^\circ
\]

Thus, we find that \( \overset{\frown}{AD} = 193^\circ \).

If you were to check if 88 was the answer, you might want to verify each step and the properties used from cyclic quadrilaterals, as \( 88^\circ \) does not align with established results. The correct final answer is:
\[
\overset{\frown}{AD} = 193^\circ
\]

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