Top Activities for Inscribed Angles - Math With Friends - Free Printable

Problem Analysis:

The task involves solving for \( x \) in each of the given geometric problems involving inscribed angles and central angles in circles. Let's solve each problem step by step.

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Problem 1 (Partner A):

#### Given:
- Central angle \( \angle AOC = 200^\circ \)
- Inscribed angle \( \angle ABC = 25x \)

#### Solution:
1. Key Property: The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
\[
\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}
\]
2. Here, the central angle \( \angle AOC = 200^\circ \), so the inscribed angle \( \angle ABC \) is:
\[
\angle ABC = \frac{1}{2} \times 200^\circ = 100^\circ
\]
3. We are given that \( \angle ABC = 25x \). Therefore:
\[
25x = 100
\]
4. Solve for \( x \):
\[
x = \frac{100}{25} = 4
\]

#### Answer for Problem 1 (Partner A):
\[
\boxed{4}
\]

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Problem 1 (Partner B):

#### Given:
- Inscribed angle \( \angle BAC = 60^\circ \)
- Central angle \( \angle BOC = 30x \)

#### Solution:
1. Key Property: The measure of a central angle is twice the measure of the inscribed angle that subtends the same arc.
\[
\text{Central angle} = 2 \times \text{Inscribed angle}
\]
2. Here, the inscribed angle \( \angle BAC = 60^\circ \), so the central angle \( \angle BOC \) is:
\[
\angle BOC = 2 \times 60^\circ = 120^\circ
\]
3. We are given that \( \angle BOC = 30x \). Therefore:
\[
30x = 120
\]
4. Solve for \( x \):
\[
x = \frac{120}{30} = 4
\]

#### Answer for Problem 1 (Partner B):
\[
\boxed{4}
\]

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Problem 2 (Partner A):

#### Given:
- Inscribed angle \( \angle BAC = 2y \)
- Inscribed angle \( \angle BDC = 8x \)

#### Solution:
1. Key Property: Inscribed angles that subtend the same arc are equal.
\[
\angle BAC = \angle BDC
\]
2. Therefore:
\[
2y = 8x
\]
3. Solve for \( y \) in terms of \( x \):
\[
y = 4x
\]

#### Answer for Problem 2 (Partner A):
Since the problem asks to solve for \( x \), and we have \( y = 4x \), we need additional information to find a numerical value for \( x \). However, based on the given information, the relationship is:
\[
\boxed{y = 4x}
\]

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Problem 2 (Partner B):

#### Given:
- Inscribed angle \( \angle BAE = 80^\circ \)
- Inscribed angle \( \angle BDE = 8x \)

#### Solution:
1. Key Property: Inscribed angles that subtend the same arc are equal.
\[
\angle BAE = \angle BDE
\]
2. Therefore:
\[
80^\circ = 8x
\]
3. Solve for \( x \):
\[
x = \frac{80}{8} = 10
\]

#### Answer for Problem 2 (Partner B):
\[
\boxed{10}
\]

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Final Answers:

1. Partner A: \( \boxed{4} \)
2. Partner B: \( \boxed{4} \)
3. Partner A: \( \boxed{y = 4x} \)
4. Partner B: \( \boxed{10} \)