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 typical problems like these. If you can describe the diagrams or provide more details (like what the angles are labeled as), I can give a more accurate solution. But here’s how such problems are usually solved:

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🔷 Key Concepts:

1. 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.
\[
\text{If } \angle AOB \text{ is central}, \text{ then } m\angle AOB = m\overset{\frown}{AB}
\]

2. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords.
- The measure of an inscribed angle is half the measure of its intercepted arc.
\[
m\angle ACB = \frac{1}{2} m\overset{\frown}{AB}
\]

3. Angle on a Semicircle: If an inscribed angle intercepts a semicircle (180° arc), it is a right angle (90°).

4. Angles subtended by the same arc: All inscribed angles that intercept the same arc are equal.

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Now, let's work through the information you provided:

> Given:
> \( m\angle AOB = 154^\circ \) → This is a central angle, so:
> \[
> m\overset{\frown}{AB} = 154^\circ
> \]

> \( m\angle ACB = 130^\circ \) → This is an inscribed angle intercepting arc \( AB \). But wait — if \( m\angle ACB = 130^\circ \), and it intercepts arc \( AB \), then:
> \[
> m\angle ACB = \frac{1}{2} m\overset{\frown}{AB} \Rightarrow 130^\circ = \frac{1}{2} m\overset{\frown}{AB}
> \]
> That would imply:
> \[
> m\overset{\frown}{AB} = 260^\circ
> \]
> But that contradicts the earlier result of \( m\overset{\frown}{AB} = 154^\circ \).

So something must be wrong with our assumption.

Wait — perhaps \( \angle ACB \) does not intercept arc \( AB \)? Let's think carefully.

Maybe point \( C \) is on the other side of the circle, so \( \angle ACB \) intercepts the major arc \( AB \), not the minor one.

But even so, inscribed angles are always half the intercepted arc.

Let’s suppose:

- \( \angle AOB = 154^\circ \) → central angle → arc \( AB = 154^\circ \)
- Then the remaining arc (the major arc \( AB \)) is:
\[
360^\circ - 154^\circ = 206^\circ
\]

Now, if \( \angle ACB = 130^\circ \), and it’s an inscribed angle, then:
\[
m\angle ACB = \frac{1}{2} \times \text{(intercepted arc)}
\Rightarrow 130^\circ = \frac{1}{2} \times \text{arc}
\Rightarrow \text{arc} = 260^\circ
\]
But 260° is greater than 360°? No — 260° is less than 360°, but we only have a total of 360°.

Wait — 2 × 130° = 260° → So the intercepted arc must be 260°.

But earlier, arc \( AB \) (minor) is 154°, so the major arc AB is 360° − 154° = 206°

So 260° ≠ 206° → contradiction.

Therefore, either the values are inconsistent, or the points are arranged differently.

Wait — maybe \( \angle ACB \) doesn’t intercept arc \( AB \)? Maybe it intercepts a different arc?

Alternatively, perhaps there’s a typo in the problem or in your description.

But you also said:
> \( m\angle ADB = 67^\circ \)

Let’s suppose that both \( \angle ACB \) and \( \angle ADB \) are inscribed angles intercepting the same arc AB.

Then they should be equal — but 130° ≠ 67°, so they cannot intercept the same arc.

So likely:

- \( \angle AOB = 154^\circ \) → central → arc \( AB = 154^\circ \)
- \( \angle ACB = 130^\circ \): This might be an inscribed angle intercepting a different arc, or possibly the reflex arc?

Wait — no. Inscribed angles are always half the intercepted arc, and the intercepted arc is the one between the two points.

But here’s a possibility: if point C is on the opposite side of the circle from O, then \( \angle ACB \) could be intercepting the major arc AB, which is 206°.

Then:
\[
m\angle ACB = \frac{1}{2} \times 206^\circ = 103^\circ
\]
But given as 130° → still not matching.

Alternatively, maybe the 130° is not an inscribed angle? Or perhaps it’s a tangent-chord angle?

Wait — let’s re-express the problem clearly.

You wrote:

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

But this seems contradictory unless the diagram shows multiple points.

Let’s suppose:

- \( O \) is the center
- \( A, B, C, D \) are points on the circle
- \( \angle AOB = 154^\circ \) → central → arc \( AB = 154^\circ \)
- \( \angle ACB \) is an inscribed angle intercepting arc \( AB \) → then:
\[
m\angle ACB = \frac{1}{2} \times 154^\circ = 77^\circ
\]
But you said \( m\angle ACB = 130^\circ \) → contradiction.

So either:
- The angle is not inscribed,
- Or it’s not intercepting arc AB,
- Or the values are mislabeled.

Alternatively, maybe \( \angle ACB = 130^\circ \) is not inscribed? But if C is on the circle, it is.

Wait — perhaps \( \angle ACB \) is not inscribed in the usual way?

Another idea: if \( \angle ACB \) is formed by chords AC and BC, and it intercepts arc AB, then it should be half of arc AB.

But 130° is too big for half of any arc in a circle (since max arc is 360°, half is 180°). So 130° is possible.

But half of 260° is 130° → so arc AB must be 260°.

But then central angle \( \angle AOB \) should be 260°, not 154°.

So unless \( \angle AOB \) is reflex, it can't be 154°.

Wait — perhaps \( \angle AOB = 154^\circ \) is the minor central angle, so arc AB = 154°, and the major arc AB is 206°.

Then an inscribed angle intercepting the major arc AB would be:
\[
\frac{1}{2} \times 206^\circ = 103^\circ
\]

Still not 130°.

So something’s off.

Wait — unless \( \angle ACB \) is not intercepting arc AB, but another arc?

Let’s suppose instead that \( \angle ACB = 130^\circ \) is not an inscribed angle intercepting arc AB, but rather, it’s part of a triangle.

Alternatively, maybe the diagram has two arcs: one minor, one major, and different points.

But without seeing the diagram, it's hard.

Let’s try to interpret the last part:

> "Find the following angles:
> a) \( m\angle ACD \)
> b) \( m\angle ABD \)
> c) \( m\angle ABC \)"

We need more info.

But you gave:

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

Let’s suppose that:

- Points A, B, C, D are on the circle
- O is center
- Arc AB = 154°
- \( \angle ACB = 130^\circ \) → inscribed → so it intercepts arc AB?
→ then \( \angle ACB = \frac{1}{2} \times \text{arc AB} = \frac{1}{2} \times 154 = 77^\circ \)
→ but given as 130° → contradiction

Unless \( \angle ACB \) intercepts the other arc — the major arc AB = 360 - 154 = 206°
→ then \( \angle ACB = \frac{1}{2} \times 206 = 103^\circ \) → still not 130°

130° → implies intercepted arc = 260°

But 260° + 154° = 414° > 360° → impossible

So the only way is if the given values are incorrect, or if the angle is not inscribed.

Wait — perhaps \( \angle ACB = 130^\circ \) is not an inscribed angle? But if C is on the circle, it is.

Another possibility: maybe \( \angle ACB \) is not the inscribed angle intercepting arc AB, but rather, it's formed by tangents or secants?

But you didn't mention tangents.

Alternatively, maybe the 130° is a typo, and it's supposed to be 77°?

Or perhaps \( m\angle ACB = 130^\circ \) is referring to a different angle.

Wait — maybe the problem says:

> "If \( m\angle AOB = 154^\circ \), \( m\angle ACB = 130^\circ \), and \( m\angle ADB = 67^\circ \)"

But perhaps \( \angle ACB \) and \( \angle ADB \) are not both inscribed angles intercepting arc AB.

Let’s suppose:

- \( \angle ADB = 67^\circ \) → inscribed angle → so it intercepts arc AB → then:
\[
m\angle ADB = \frac{1}{2} \times m\overset{\frown}{AB} \Rightarrow 67^\circ = \frac{1}{2} \times m\overset{\frown}{AB} \Rightarrow m\overset{\frown}{AB} = 134^\circ
\]
But earlier, \( m\angle AOB = 154^\circ \) → arc AB = 154° → contradiction.

So again, inconsistency.

Therefore, the values provided are incompatible unless:

- One of the angles is not inscribed,
- Or the arc is not AB,
- Or the diagram has multiple arcs.

Perhaps \( \angle ACB = 130^\circ \) is not inscribed, but is a triangle angle?

Wait — maybe the diagram shows triangle ABC inscribed in the circle, and \( \angle ACB = 130^\circ \), but then the arc AB would be 260°, which means central angle is 260°, but you said 154° — impossible.

So unless the 154° is the reflex angle, but typically central angles are taken as smaller ones.

Alternatively, maybe \( m\angle AOB = 154^\circ \) is the minor arc, and \( \angle ACB \) is not intercepting that arc.

Let’s suppose:

- Arc AB = 154° (minor)
- Then major arc AB = 206°
- If point C is on the major arc, then \( \angle ACB \) intercepts arc AB (minor), so:
\[
m\angle ACB = \frac{1}{2} \times 154^\circ = 77^\circ
\]
- If point C is on the minor arc, then \( \angle ACB \) would be intercepting the major arc, so:
\[
m\angle ACB = \frac{1}{2} \times 206^\circ = 103^\circ
\]
- But you said 130° — still not matching.

So none of these match.

Similarly, \( \angle ADB = 67^\circ \) → if D is on the circle, and it intercepts arc AB, then:
\[
67^\circ = \frac{1}{2} \times \text{arc} \Rightarrow \text{arc} = 134^\circ
\]

But arc AB is 154° → so D must be intercepting a different arc.

So perhaps \( \angle ADB \) intercepts arc AB, but then arc AB must be 134°, contradicting 154°.

So unless the values are for different arcs.

Let’s assume that:

- \( m\angle AOB = 154^\circ \) → arc AB = 154°
- \( \angle ADB = 67^\circ \) → inscribed angle intercepting arc AB → then:
\[
\angle ADB = \frac{1}{2} \times 154^\circ = 77^\circ
\]
But given as 67° → close but not exact.

So perhaps the numbers are approximate, or there's a mistake.

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

Let’s suppose that \( \angle ADB \) intercepts arc AC or something else.

Without the diagram, it's very hard.

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✔ Conclusion:

There is likely a contradiction in the given values:

- If \( m\angle AOB = 154^\circ \), then arc AB = 154°
- Then any inscribed angle intercepting arc AB should be \( 77^\circ \)
- But you say \( m\angle ACB = 130^\circ \) and \( m\angle ADB = 67^\circ \) — neither matches 77°

So either:
- The values are incorrect,
- Or the angles are not intercepting arc AB,
- Or the diagram has additional elements (like tangents, chords, etc.)

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🛠️ What You Should Do:

Please describe the diagram or clarify:

1. Where are points A, B, C, D located?
2. Is O the center?
3. Which arcs do the angles intercept?
4. Are all angles inscribed or some are central?

Alternatively, if you can type out the problems from the image, I can solve them accurately.

For example, if the first problem is:

> In a circle, \( \angle AOB = 154^\circ \), find \( m\angle ACB \) where C is on the circle.

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

But since you said \( m\angle ACB = 130^\circ \), it’s likely a different configuration.

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Final Request:

Please provide a text description of the image or the exact problem statements. For example:

- "In circle O, arc AB = 154°, point C is on the circle, find \( m\angle ACB \)"
- "Given \( m\angle AOB = 154^\circ \), \( m\angle ACB = 130^\circ \), find \( m\angle ACD \)" — etc.

Once I know the actual setup, I can give the correct solution.

Until then, I can only speculate, and the values you provided seem inconsistent.