Inscribed Angles and Central Angles Worksheets - Free Printable

Since I can't view or access images directly, I can't see the specific details of your image. However, you've mentioned that it's about central and inscribed angles in circles, and you're asked to:

1. Find the value of \( x \) and the measure of the indicated angle in each circle.
2. Use the fact that \( m\angle AOB = 154^\circ \), \( m\angle ACB = 130^\circ \), and \( m\angle ADB = 67^\circ \) to find certain angles.

Let me help you solve this step-by-step based on common problems like these. If you can describe the diagrams or provide more details (e.g., what arcs or angles are labeled), I can tailor the solution better.

But for now, here’s a general explanation of how to solve central and inscribed angle problems, followed by an example based on typical configurations.

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🔷 Key Concepts:

- Central Angle: An angle whose vertex is at the center of the circle. The measure of a central angle is equal to the measure of its intercepted arc.

\[
\text{If } \angle AOB \text{ is central, then } m\angle AOB = m\widehat{AB}
\]

- Inscribed Angle: An angle whose vertex is on the circle and whose sides intersect the circle. The measure of an inscribed angle is half the measure of its intercepted arc.

\[
m\angle ACB = \frac{1}{2} m\widehat{AB}
\]

- Angles subtended by the same arc: All inscribed angles that intercept the same arc are congruent.

- Angle in a semicircle: An inscribed angle that intercepts a semicircle is a right angle (90°).

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✔ Problem 1: Finding \( x \) and the indicated angle

Let’s suppose the diagram shows a circle with various angles labeled. Here are common types of problems:

#### Example 1: Inscribed Angle
Suppose \( \angle ABC \) is an inscribed angle intercepting arc \( AC \), and \( m\angle ABC = x \), with arc \( AC = 80^\circ \).

Then:
\[
x = \frac{1}{2} \times 80^\circ = 40^\circ
\]

#### Example 2: Central Angle
If \( \angle AOC \) is a central angle and \( m\angle AOC = 120^\circ \), then arc \( AC = 120^\circ \).

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✔ Problem 2: Given:

- \( m\angle AOB = 154^\circ \) → This is a central angle, so arc \( AB = 154^\circ \)
- \( m\angle ACB = 130^\circ \) → This is an inscribed angle
- \( m\angle ADB = 67^\circ \) → Another inscribed angle

Let’s analyze:

#### Step 1: Use \( m\angle ACB = 130^\circ \)
This is an inscribed angle intercepting arc \( AB \). But wait — if arc \( AB = 154^\circ \), then the inscribed angle should be half of that:

\[
m\angle ACB = \frac{1}{2} \times 154^\circ = 77^\circ
\]

But you’re told \( m\angle ACB = 130^\circ \), which contradicts unless it’s intercepting a different arc.

Wait — this suggests that point \( C \) is on the opposite side of the circle from the minor arc \( AB \). That means \( \angle ACB \) intercepts the major arc \( AB \), not the minor one.

So:

- Minor arc \( AB = 154^\circ \)
- Major arc \( AB = 360^\circ - 154^\circ = 206^\circ \)

Then, if \( \angle ACB \) is inscribed and intercepts major arc \( AB \), we have:

\[
m\angle ACB = \frac{1}{2} \times \text{major arc } AB = \frac{1}{2} \times 206^\circ = 103^\circ
\]

But you said \( m\angle ACB = 130^\circ \), so this doesn’t match either.

Wait — something’s off.

Let’s double-check.

Alternatively, perhaps \( \angle ACB \) is not intercepting arc \( AB \), but another arc.

Let me try a different interpretation.

Suppose:
- \( \angle AOB = 154^\circ \): central angle → arc \( AB = 154^\circ \)
- \( \angle ACB = 130^\circ \): inscribed angle → must intercept an arc of \( 2 \times 130^\circ = 260^\circ \)
- So arc \( AB \) cannot be 154° if it's being intercepted by an inscribed angle of 130°.

This suggests that \( \angle ACB \) does not intercept arc \( AB \), but rather some other arc.

Perhaps the points are arranged such that \( C \) is on the other side, and \( \angle ACB \) intercepts arc \( AB \) not including the center.

Wait — actually, an inscribed angle cannot be larger than 180°, and it’s always half the intercepted arc.

So if \( \angle ACB = 130^\circ \), then the arc it intercepts is:

\[
\text{arc} = 2 \times 130^\circ = 260^\circ
\]

That’s possible — it would be the major arc.

So if arc \( AB = 260^\circ \), then central angle \( \angle AOB = 260^\circ \), but you said \( m\angle AOB = 154^\circ \), so again contradiction.

Wait — this means the two angles cannot both be referring to the same arc.

So likely, the problem has multiple points and arcs.

Let’s assume the following configuration:

- Points \( A, B, C, D \) lie on the circle.
- \( O \) is the center.
- \( \angle AOB = 154^\circ \) → so minor arc \( AB = 154^\circ \)
- \( \angle ACB = 130^\circ \) → inscribed angle
- \( \angle ADB = 67^\circ \) → another inscribed angle

Now, since \( \angle ACB = 130^\circ \), and inscribed angles are less than 180°, it's possible.

But it must intercept an arc of \( 2 \times 130^\circ = 260^\circ \), so it must be intercepting the major arc between \( A \) and \( B \)? But that would be \( 360^\circ - 154^\circ = 206^\circ \), which is not 260°.

Still inconsistent.

Wait — maybe \( \angle ACB \) intercepts arc \( AB \), but \( C \) is on the minor arc side, so it intercepts the minor arc? Then:

\[
m\angle ACB = \frac{1}{2} \times 154^\circ = 77^\circ
\]

But given as 130°, so no.

Alternatively, perhaps \( \angle ACB \) is not intercepting arc \( AB \), but arc \( AD \) or something else.

Let’s consider that \( \angle ADB = 67^\circ \). Since it's inscribed, it intercepts arc \( AB \), so:

\[
m\angle ADB = \frac{1}{2} \times \text{arc } AB \Rightarrow 67^\circ = \frac{1}{2} \times \text{arc } AB \Rightarrow \text{arc } AB = 134^\circ
\]

But earlier, \( m\angle AOB = 154^\circ \), which implies arc \( AB = 154^\circ \), contradiction.

So now we have a conflict.

Therefore, the only way this makes sense is if:

- \( m\angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- Then any inscribed angle intercepting arc \( AB \) should be \( \frac{1}{2} \times 154^\circ = 77^\circ \)

But you say \( m\angle ACB = 130^\circ \) and \( m\angle ADB = 67^\circ \)

So let’s suppose:

- \( \angle ADB = 67^\circ \) → inscribed angle intercepting arc \( AB \)
→ then arc \( AB = 2 \times 67^\circ = 134^\circ \)

But \( \angle AOB = 154^\circ \), so arc \( AB = 154^\circ \)

Contradiction.

So unless \( \angle ADB \) is not intercepting arc \( AB \), this can't work.

Wait — maybe \( \angle ADB \) intercepts arc \( AB \), but \( D \) is on the other side, so it sees the major arc?

No — inscribed angle always sees the arc between the two points, and it's the minor arc unless specified.

But if \( D \) is on the major arc, then \( \angle ADB \) intercepts arc \( AB \), and its measure is half the arc.

So if \( \angle ADB = 67^\circ \), then arc \( AB = 134^\circ \)

But \( \angle AOB = 154^\circ \), so arc \( AB = 154^\circ \)

Conflict.

Unless... the problem says:

> If \( m\angle AOB = 154^\circ \), \( m\angle ACB = 130^\circ \), \( m\angle ADB = 67^\circ \), find the following angles.

But that can’t all be true unless the arcs are different.

Wait — perhaps \( \angle ACB \) and \( \angle ADB \) are not both intercepting arc \( AB \).

Maybe:

- \( \angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- \( \angle ADB = 67^\circ \) → inscribed angle intercepting arc \( AB \) → then \( \angle ADB = \frac{1}{2} \times 154^\circ = 77^\circ \), but given as 67° → no.

Alternatively, maybe \( \angle ADB \) intercepts a different arc.

Let’s suppose the circle has points \( A, B, C, D \), and \( O \) is center.

Let’s suppose:
- \( \angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- \( \angle ACB = 130^\circ \) → inscribed angle, so it intercepts an arc of \( 2 \times 130^\circ = 260^\circ \)
- So arc \( AB \) must be 260°? But that contradicts 154°.

So unless \( \angle ACB \) is not intercepting arc \( AB \), but arc \( AD \) or \( BD \), etc.

Wait — perhaps there’s a typo in the problem, or in my understanding.

Alternatively, maybe \( \angle ACB \) is not an inscribed angle intercepting arc \( AB \), but rather a cyclic quadrilateral angle.

But without a diagram, it's hard.

---

Let’s try to interpret the most likely scenario:

Assume the problem is asking:

Given:
- \( m\angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- \( m\angle ACB = 130^\circ \) → inscribed angle
- \( m\angle ADB = 67^\circ \) → inscribed angle

Find:
a) \( m\angle ACD \)
b) \( m\angle ABD \)
c) \( m\angle ADC \)

But still, need more info.

Wait — perhaps \( \angle ACB = 130^\circ \) is not related to arc \( AB \), but to arc \( AB \) via a different path.

Another idea: The measure of an inscribed angle is half the arc it intercepts.

So if \( \angle ACB = 130^\circ \), then the arc it intercepts is \( 260^\circ \), which is impossible unless it's the major arc.

But the total circle is 360°, so arc \( AB \) could be 260°, then central angle \( \angle AOB = 260^\circ \), but you said 154°, so no.

Wait — unless \( \angle AOB = 154^\circ \) is the minor arc, and \( \angle ACB = 130^\circ \) is an inscribed angle intercepting the major arc.

But then:

- minor arc \( AB = 154^\circ \)
- major arc \( AB = 360^\circ - 154^\circ = 206^\circ \)
- inscribed angle intercepting major arc \( AB \) would be \( \frac{1}{2} \times 206^\circ = 103^\circ \)

But you said \( \angle ACB = 130^\circ \), so not matching.

Now, \( \angle ADB = 67^\circ \) — if it intercepts minor arc \( AB \), then:

\[
\angle ADB = \frac{1}{2} \times 154^\circ = 77^\circ
\]

But given as 67° — close but not exact.

Perhaps there’s a typo.

Alternatively, maybe the given values are not all for the same arc.

Let’s suppose:

- \( m\angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- \( m\angle ACB = 130^\circ \) → this might be an angle at \( C \) formed by chords \( AC \) and \( BC \), intercepting arc \( AB \), but that would be half of arc \( AB \) = 77°, not 130°.

Wait — unless \( \angle ACB \) is not an inscribed angle, but a central angle? No, it's labeled as \( \angle ACB \), so vertex at \( C \), on the circle.

So it must be inscribed.

But 130° > 90°, so possible.

But it must intercept an arc of 260°.

So arc \( AB = 260^\circ \), then central angle \( \angle AOB = 260^\circ \), but you said 154°.

So inconsistency.

Unless the problem has a typo.

Alternatively, maybe the given values are for different diagrams.

Perhaps the "if" statement is not part of the diagram, but a separate problem.

Let’s look at the structure:

> 6) If \( m\angle AOB = 154^\circ \), \( m\angle ACB = 130^\circ \), \( m\angle ADB = 67^\circ \), find the following angles:

But this is impossible unless the arcs are different.

Wait — perhaps \( \angle ACB = 130^\circ \) is not an inscribed angle, but an angle inside the circle, like a triangle.

But still, it's likely inscribed.

Another possibility: the points are arranged so that \( \angle ACB \) intercepts arc \( AB \), but it's a reflex angle? No, inscribed angles are less than 180°.

I think there might be a mistake in the problem or in the values.

Alternatively, maybe \( \angle ACB = 130^\circ \) is not an inscribed angle, but a triangle angle.

Let’s assume the following:

- \( \angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- Then any inscribed angle intercepting arc \( AB \) is \( 77^\circ \)
- So if \( \angle ADB = 67^\circ \), it must be intercepting a different arc.

Let’s suppose:

- \( \angle ADB = 67^\circ \) → inscribed angle → intercepts arc \( AB \) → then arc \( AB = 134^\circ \)
- But \( \angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)

Contradiction.

So unless the two statements are for different circles or different parts, they can't both be true.

Therefore, I suspect there may be a typo in the problem.

---

Most Likely Scenario:

Perhaps the values are meant to be consistent.

Let’s assume:

- \( m\angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- Then inscribed angle intercepting arc \( AB \) is \( 77^\circ \)
- So if \( \angle ADB = 67^\circ \), it might be intercepting a different arc, say arc \( AD \) or \( BD \)

But without a diagram, it's hard.

Alternatively, maybe the problem is:

Given:
- \( m\angle AOB = 154^\circ \)
- \( m\angle ACB = 130^\circ \) — this might be a typo; perhaps it's 77°

Or perhaps \( m\angle ACB = 130^\circ \) is the measure of an angle at \( C \) that is not inscribed, but part of a triangle.

But I think the best approach is to assume the standard rules and solve for typical problems.

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Let’s solve a typical problem:

#### Problem: In a circle, \( \angle AOB = 154^\circ \). Find the measure of inscribed angle \( \angle ACB \) that intercepts arc \( AB \).

Solution:
\[
m\angle ACB = \frac{1}{2} \times m\widehat{AB} = \frac{1}{2} \times 154^\circ = 77^\circ
\]

#### If \( \angle ADB = 67^\circ \), and it's inscribed, then:
\[
m\widehat{AB} = 2 \times 67^\circ = 134^\circ
\]

So if both are true, arc \( AB \) can't be both 154° and 134°.

So likely, the given values are for different parts of the diagram.

Perhaps the "if" statement is for a different diagram.

Given the confusion, I recommend checking the diagram carefully.

But based on standard problems, here’s how to solve:

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Final Answer (based on common problems):

Assume the following:

- In diagram 6, \( m\angle AOB = 154^\circ \) → arc \( AB = 154^\circ \)
- \( \angle ACB \) is inscribed, intercepting arc \( AB \) → \( m\angle ACB = \frac{1}{2} \times 154^\circ = 77^\circ \)
- But given as 130° — so perhaps it's a different angle.

Alternatively, if \( \angle ACB = 130^\circ \), then arc it intercepts is 260°, so it's the major arc.

Then minor arc \( AB = 360^\circ - 260^\circ = 100^\circ \), so central angle \( \angle AOB = 100^\circ \), not 154°.

So inconsistency.

Therefore, I suspect a typo in the problem.

But if you can describe the diagrams or provide the text of the problems, I can give accurate answers.

For now, here’s a general rule:

> Inscribed angle = ½ × intercepted arc
> Central angle = intercepted arc

Use that to solve.

If you can describe the first few diagrams (e.g., "circle with central angle 60°, find inscribed angle"), I can solve them step by step.

Please provide more details or describe the diagrams!