Let me help you analyze the worksheet questions based on the image. Here's a breakdown of the problems and their solutions:

1. Which inscribed angles are congruent?

Solution: Inscribed angles subtended by the same arc in a circle are congruent. Based on the diagram:

• $\angle BCD$ and $\angle BAD$ both subtend arc $BD$, so they are congruent.

Thus, the congruent angles are $\angle BCD$ and $\angle BAD$.

2. If $m \angle CBD = 54^\circ$, what is the measure of $\overset{\frown}{CD}$?

Solution: In a circle, the inscribed angle is half the measure of the intercepted arc. Thus: $m \overset{\frown}{CD} = 2 \times m \angle CBD = 2 \times 54^\circ = 108^\circ$

3. If $m \overset{\frown}{AB} = 96^\circ$, what is the measure of $\angle ACB$?

Solution: The measure of an inscribed angle is half of the arc it intercepts. Since $\angle ACB$ intercepts arc $AB$, we have: $m \angle ACB = \frac{1}{2} \times m \overset{\frown}{AB} = \frac{1}{2} \times 96^\circ = 48^\circ$

4. If $m \angle ABD = 5x + 3$ and $m \angle DCA = 4x + 10$, find:

a. The value of $x$

Since both angles subtend the same arc $AD$, they must be equal. Therefore, we set the expressions for the angles equal to each other: $5x + 3 = 4x + 10$ Solving for $x$: $x = 7$

b. $m \angle ABD$

Substitute $x = 7$ into $5x + 3$: $m \angle ABD = 5(7) + 3 = 35 + 3 = 38^\circ$

c. $m \angle DCA$

Since $m \angle ABD = m \angle DCA$ (as they subtend the same arc), $m \angle DCA = 38^\circ$.

d. $m \overset{\frown}{AD}$

Since both angles subtend arc $AD$, we know: $m \overset{\frown}{AD} = 2 \times m \angle ABD = 2 \times 38^\circ = 76^\circ$

5. If $m \angle BDC = 6x - 4$ and $m \overset{\frown}{BC} = 10x + 2$, find:

a. The value of $x$

The angle $\angle BDC$ is an inscribed angle subtending arc $BC$, so: $m \overset{\frown}{BC} = 2 \times m \angle BDC$ Substitute the expressions: $10x + 2 = 2 \times (6x - 4)$ $10x + 2 = 12x - 8$ Solving for $x$: $2x = 10 \quad \Rightarrow \quad x = 5$

b. $m \angle BDC$

Substitute $x = 5$ into $6x - 4$: $m \angle BDC = 6(5) - 4 = 30 - 4 = 26^\circ$

c. $m \overset{\frown}{BC}$

Substitute $x = 5$ into $10x + 2$: $m \overset{\frown}{BC} = 10(5) + 2 = 50 + 2 = 52^\circ$

d. $m \angle BAC$

Since $m \angle BAC$ subtends arc $BC$, we can use the inscribed angle rule: $m \angle BAC = \frac{1}{2} \times m \overset{\frown}{BC} = \frac{1}{2} \times 52^\circ = 26^\circ$

Would you like more details or explanations on any of these solutions?

Here are 5 additional questions for deeper understanding:

1. Why are the inscribed angles subtending the same arc always congruent?
2. How do we determine the measure of an arc from an inscribed angle?
3. What is the relationship between the arc and its central angle in a circle?
4. Can two inscribed angles subtend different arcs yet still be congruent?
5. How does the circle theorem apply when calculating the measure of an intercepted arc?

Tip: Always remember that an inscribed angle is half the measure of the intercepted arc.