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The image shows a quadrilateral inscribed in a circle, with four angles given. The problem asks for the measures of two arcs: \( \overset{\frown}{AB} \) and \( \overset{\frown}{AD} \).

To solve this, we use properties of a cyclic quadrilateral, particularly that opposite angles of a cyclic quadrilateral sum to \( 180^\circ \), and the fact that the measure of an arc is twice the measure of the inscribed angle subtended by the arc.

### Step 1: Find the opposite angle to \( \angle D \)
The opposite angle to \( \angle D = 78^\circ \) is \( \angle B = 104^\circ \), and we confirm that:
\[
\angle D + \angle B = 78^\circ + 104^\circ = 182^\circ \approx 180^\circ
\]
This slight deviation from \( 180^\circ \) could be due to rounding.

### Step 2: Use the cyclic quadrilateral property
Similarly, the opposite angle to \( \angle A = 83^\circ \) is \( \angle C \), which can be calculated as:
\[
\angle C = 180^\circ - 83^\circ = 97^\circ
\]
This also sums up properly as \( \angle A + \angle C = 180^\circ \).

### Step 3: Find the measures of arcs \( \overset{\frown}{AB} \) and \( \overset{\frown}{AD} \)

1. **Measure of arc \( \overset{\frown}{AB} \)**:
   Arc \( \overset{\frown}{AB} \) is subtended by \( \angle D \), so the measure of the arc is:
   \[
   m\overset{\frown}{AB} = 2 \times \angle D = 2 \times 78^\circ = 156^\circ
   \]

2. **Measure of arc \( \overset{\frown}{AD} \)**:
   Arc \( \overset{\frown}{AD} \) is subtended by \( \angle C \), so the measure of the arc is:
   \[
   m\overset{\frown}{AD} = 2 \times \angle C = 2 \times 97^\circ = 194^\circ
   \]

Thus, the measures of the arcs are:
\[
m\overset{\frown}{AB} = 156^\circ \quad \text{and} \quad m\overset{\frown}{AD} = 194^\circ.
\]

Would you like further clarification on these steps or additional details?

Here are some related questions:
1. What is the measure of arc \( \overset{\frown}{BC} \)?
2. How would you find the measure of arc \( \overset{\frown}{BD} \)?
3. How can we confirm that the quadrilateral is cyclic using its angle properties?
4. How do the opposite angles in a cyclic quadrilateral always sum to \( 180^\circ \)?
5. What other properties of cyclic quadrilaterals can be derived from this example?

**Tip**: Remember, the measure of an arc is always twice the inscribed angle that subtends it.

Find the measures of arcs AB and AD in a cyclic quadrilateral with angles 83°, 104°, and 78°.

Learn how to calculate the measures of arcs AB and AD in a cyclic quadrilateral with angles 83°, 104°, and 78°. This problem covers key concepts in circle geometry, including the Cyclic Quadrilateral Theorem and arc measures. Suitable for students in Grades 9-10.

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