Free Collection of Inscribed Angles Worksheets for Students - Free Printable

Since I can't view or access uploaded images directly, I can’t see the specific diagram or details of your image. However, based on the text you've provided, it appears to be a geometry worksheet focused on arcs, central angles, and inscribed angles in circles.

Let me help you solve this step-by-step using the information given in your text:

---

Given:

- $ m\angle CPX = 28^\circ $
- $ m\angle CPY = 38^\circ $
- $ WZ $ and $ XV $ are diameters.
- $ O $ is the center of the circle in all figures.

We are asked to find various arc measures and angle measures, particularly related to arcs and angles in circles.

Let’s go through each problem systematically.

---

Key Concepts (Review):

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{Measure of arc} = \text{Measure of central angle} $

2. 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.

> $ \text{Inscribed angle} = \frac{1}{2} \times \text{intercepted arc} $

3. Diameter: A chord passing through the center. It divides the circle into two semicircles (each 180°).

4. Vertical Angles: Equal in measure.

5. Straight Angle: 180° — useful when dealing with diameters.

6. Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of their measures.

---

Now let's solve each question based on the given info.

---

Problem 1: $ \overarc{PZ} $

We know:
- $ \angle CPX = 28^\circ $
- $ \angle CPY = 38^\circ $
- $ XV $ and $ WZ $ are diameters → so they pass through the center $ O $

Assuming point $ C $ is on the circle, and $ P $ is the center? Wait — but the notation uses $ O $ as the center.

Wait — in the diagram, $ O $ is labeled as the center, so $ P $ may not be the center.

But we have angles like $ \angle CPX $, $ \angle CPY $. Let's interpret carefully.

Let’s assume that $ C $, $ P $, $ X $, $ Y $, etc., are points on the circle or inside.

But if $ \angle CPX $ and $ \angle CPY $ are given, and $ P $ is not the center, then these might be inscribed angles?

Wait — but the notation $ m\angle CPX = 28^\circ $ suggests that $ P $ is the vertex.

But $ O $ is the center. So likely $ P $ is a point on the circle.

But if $ \angle CPX $ is at point $ P $, and $ C $, $ P $, $ X $ are on the circle, then $ \angle CPX $ is an inscribed angle.

But wait — the measure of $ \angle CPX = 28^\circ $, and $ \angle CPY = 38^\circ $. Both share point $ P $, and rays going to $ C $, $ X $, $ Y $.

So perhaps $ P $ is a point on the circle, and $ C $, $ X $, $ Y $ are other points.

But then we need more context.

Alternatively, maybe $ P $ is the center? But $ O $ is already labeled as the center.

This is confusing.

Wait — in many problems, $ O $ is the center, and $ P $ is another point.

But here, $ \angle CPX $ and $ \angle CPY $ are given, and both involve point $ P $.

Let’s suppose that $ P $ is the center. But the problem says $ O $ is the center.

So unless $ P = O $, which seems unlikely.

Wait — look at the first few questions:

> Find each measure in $ \odot P $ if $ m\angle CPX = 28^\circ $, $ m\angle CPY = 38^\circ $, and $ WZ $, $ XV $ are diameters.

Ah! Here’s the key: the circle is named $ \odot P $ — meaning P is the center.

So even though later diagrams show $ O $ as the center, in this section, P is the center.

So $ P $ is the center of the circle.

That makes sense now.

So:

- $ \angle CPX = 28^\circ $: this is a central angle, since $ P $ is the center.
- $ \angle CPY = 38^\circ $: also a central angle.
- $ WZ $ and $ XV $ are diameters → pass through center $ P $.

So now we can proceed.

---

Step 1: Use given angles

Given:
- $ \angle CPX = 28^\circ $ → central angle → intercepts arc $ \overarc{CX} $
- So $ m\overarc{CX} = 28^\circ $

Similarly:
- $ \angle CPY = 38^\circ $ → intercepts arc $ \overarc{CY} $
- So $ m\overarc{CY} = 38^\circ $

Now, depending on the position of points, we can find other arcs.

But we need to figure out the configuration.

Let’s assume points are arranged around the circle.

Suppose from point $ C $, we go to $ X $ and $ Y $, and $ \angle CPX = 28^\circ $, $ \angle CPY = 38^\circ $. Since both angles share ray $ PC $, and differ in direction, probably $ X $ and $ Y $ are on different sides.

But without a diagram, we need to make logical assumptions.

But wait — there’s a diagram implied.

Let’s try to reconstruct.

Let’s suppose:
- $ P $ is the center.
- $ WZ $ and $ XV $ are diameters → so $ W $ and $ Z $ are endpoints of a diameter, $ X $ and $ V $ are endpoints of another diameter.
- So $ WZ $ passes through $ P $, $ XV $ passes through $ P $.
- So $ W $, $ P $, $ Z $ are colinear; $ X $, $ P $, $ V $ are colinear.

Now, $ \angle CPX = 28^\circ $: angle at center $ P $ between points $ C $, $ P $, $ X $. So $ C $ is some point on the circle.

Similarly, $ \angle CPY = 38^\circ $: angle between $ C $, $ P $, $ Y $.

So point $ C $ is somewhere on the circle, and we have two angles from $ C $ to $ X $ and $ C $ to $ Y $.

But we don’t know where $ Y $ is.

Wait — perhaps $ Y $ is on the same side as $ X $? Or opposite?

But $ XV $ is a diameter, so $ V $ is opposite $ X $.

Maybe $ Y $ is on the circle, and $ \angle CPY = 38^\circ $.

But unless we know how $ X $, $ Y $, $ C $ are positioned, we can't determine arcs.

But let’s look at the questions.

---

Question 1: $ \overarc{PZ} $

Wait — $ \overarc{PZ} $? That would be an arc from $ P $ to $ Z $. But $ P $ is the center — not on the circle.

So arc $ \overarc{PZ} $ doesn’t make sense — arcs are between points on the circle.

Likely typo or mislabeling.

Possibly meant $ \overarc{WZ} $? But $ WZ $ is a diameter — so $ \overarc{WZ} $ is a semicircle → 180°.

But question says $ \overarc{PZ} $ — invalid.

Alternatively, maybe it's $ \overarc{CZ} $ or $ \overarc{ZX} $?

Wait — let's re-express.

Perhaps $ P $ is the center, and $ Z $ is on the circle — so $ \overarc{PZ} $ is meaningless.

More likely, the arc is from $ C $ to $ Z $, or $ X $ to $ Z $, etc.

But the notation is $ \overarc{PZ} $ — this is problematic.

Unless "P" is not the center? But earlier it says $ \odot P $, so P is the center.

Wait — maybe it's a typo, and it should be $ \overarc{CZ} $ or $ \overarc{WX} $?

Alternatively, perhaps $ P $ is on the circle? But then $ \odot P $ wouldn't make sense.

I think there's confusion.

Wait — in the list, the first few questions are:

1. $ \overarc{PZ} $
2. $ \overarc{WK} $
3. $ \angle NPZ $
4. $ \overarc{VWZ} $
5. $ \angle XPY $
6. $ \overarc{XY} $
7. $ \overarc{WZ} $

Hmm — $ \overarc{PZ} $ again.

But $ P $ is the center — so it can't be an endpoint of an arc.

So likely, this is a mistake, or perhaps $ P $ is on the circle in some diagrams.

But earlier it says $ \odot P $, so P is the center.

Alternatively, perhaps the circle is $ \odot O $, and P is a point on the circle.

But the problem says: “Find each measure in $ \odot P $” — so circle with center P.

So P is the center.

Then $ \overarc{PZ} $ is impossible.

So likely, it's a typo, and it should be $ \overarc{CZ} $, $ \overarc{XZ} $, etc.

Alternatively, maybe $ \overarc{PZ} $ means arc from $ C $ to $ Z $, but labeled wrong?

This is ambiguous.

But let’s skip to the ones we can do.

---

Question 5: $ \angle XPY $

We are told:
- $ \angle CPX = 28^\circ $
- $ \angle CPY = 38^\circ $

Both angles share ray $ PC $, and go to $ X $ and $ Y $.

So if $ X $ and $ Y $ are on the same side of $ PC $, then $ \angle XPY = |\angle CPY - \angle CPX| = |38^\circ - 28^\circ| = 10^\circ $

If they are on opposite sides, then $ \angle XPY = 28^\circ + 38^\circ = 66^\circ $

But we need to know the configuration.

But notice: $ XV $ is a diameter. So $ X $ and $ V $ are opposite.

Also, $ WZ $ is a diameter.

Now, $ \angle CPX = 28^\circ $: central angle from $ C $ to $ X $

$ \angle CPY = 38^\circ $: from $ C $ to $ Y $

So if $ Y $ is on the same side as $ X $ from $ C $, then $ \angle XPY = 38^\circ - 28^\circ = 10^\circ $

But if $ Y $ is on the other side, then $ 38^\circ + 28^\circ = 66^\circ $

But without diagram, hard to say.

But perhaps from the names: $ \angle CPY $, $ \angle CPX $, and $ XV $ is a diameter, so $ X $ and $ V $ are opposite.

Maybe $ Y $ is near $ X $.

But let’s assume $ X $ and $ Y $ are on the same side of $ PC $, and $ \angle CPY > \angle CPX $, so $ Y $ is further from $ C $ than $ X $.

Then $ \angle XPY = \angle CPY - \angle CPX = 38^\circ - 28^\circ = 10^\circ $

So answer: $ \angle XPY = 10^\circ $

But we need to confirm.

Alternatively, maybe $ \angle XPY $ is the angle at $ P $ between $ X $ and $ Y $, which could be the sum or difference.

But without diagram, best guess: $ \angle XPY = 10^\circ $

But let’s hold off.

---

Question 6: $ \overarc{XY} $

We want arc $ XY $. This is intercepted by central angle $ \angle XPY $.

So $ m\overarc{XY} = m\angle XPY $

From above, if $ \angle XPY = 10^\circ $, then $ \overarc{XY} = 10^\circ $

But we need to verify.

Alternatively, if $ \angle XPY = 66^\circ $, then arc is 66°.

But let’s see other clues.

We also have $ \angle CPX = 28^\circ $, so arc $ CX = 28^\circ $

$ \angle CPY = 38^\circ $, so arc $ CY = 38^\circ $

So if $ X $ and $ Y $ are on the same side of $ C $, then arc $ XY = |38^\circ - 28^\circ| = 10^\circ $

If on opposite sides, arc $ XY = 38^\circ + 28^\circ = 66^\circ $

But arc $ XY $ could be minor or major.

But likely minor arc.

So probably $ \overarc{XY} = 10^\circ $

So for Q6: $ \overarc{XY} = 10^\circ $

Then Q5: $ \angle XPY = 10^\circ $

---

Question 7: $ \overarc{WZ} $

$ WZ $ is a diameter → so arc $ WZ $ is a semicircle → $ 180^\circ $

Answer: $ 180^\circ $

---

Question 4: $ \overarc{VWZ} $

This is arc from $ V $ to $ W $ to $ Z $ — so it goes from $ V $ to $ Z $ passing through $ W $

Since $ WV $ and $ WZ $ are parts of the circle.

But $ XV $ is a diameter, so $ X $ and $ V $ are endpoints.

$ WZ $ is a diameter, so $ W $ and $ Z $ are endpoints.

So points $ V $, $ W $, $ Z $ are on the circle.

Arc $ VWZ $ is from $ V $ to $ Z $ via $ W $.

So it's the arc from $ V $ to $ Z $ passing through $ W $.

The total circle is 360°.

But we need to know the position.

But since $ XV $ and $ WZ $ are diameters, they intersect at $ P $, the center.

So the two diameters cross at $ P $, forming four angles.

Let’s suppose $ \angle CPX = 28^\circ $, $ \angle CPY = 38^\circ $

But we don't know where $ C $ is.

Perhaps $ C $ is at $ W $ or $ V $?

Without diagram, very hard.

But let’s move to the next set.

---

Problems 8–20: Diagrams with $ O $ as center

Now the instructions change:

> In each of the following figures, $ O $ is the center of the circle. Calculate the value of $ x $ and justify your answer.

And there are diagrams with angles and arcs labeled.

Since I can't see the diagrams, I can only give general guidance.

But let’s assume typical problems.

For example:

#### Problem 8: A circle with center $ O $, and an inscribed angle $ x $ intercepting an arc of $ 100^\circ $

Then $ x = \frac{1}{2} \times 100^\circ = 50^\circ $

#### Problem 9: Central angle $ x $ intercepting arc $ 100^\circ $ → $ x = 100^\circ $

#### Problem 10: Inscribed angle $ x $ intercepting arc $ 100^\circ $ → $ x = 50^\circ $

#### Problem 11: Two chords intersecting, forming angle $ x $, with intercepted arcs $ a $ and $ b $ → $ x = \frac{1}{2}(a + b) $

#### Problem 12: Tangent-chord angle → $ x = \frac{1}{2} \times \text{intercepted arc} $

#### Problem 13: Triangle with vertices on circle, one angle given, find $ x $

#### Problem 14: Isosceles triangle with two radii, base angles equal

#### Problem 15: Arc $ 120^\circ $, inscribed angle $ x $ → $ x = 60^\circ $

#### Problem 16: Central angle $ 120^\circ $, arc $ x $ → $ x = 120^\circ $

#### Problem 17: Right triangle inscribed in semicircle → $ x = 90^\circ $

#### Problem 18: Arc $ 100^\circ $, inscribed angle $ x $ → $ x = 50^\circ $

#### Problem 19: Arc $ 100^\circ $, central angle $ x $ → $ x = 100^\circ $

#### Problem 20: Two arcs: $ 120^\circ $ and $ 101^\circ $, find $ x $

Wait — Problem 20 shows arc $ 120^\circ $, arc $ 101^\circ $, and angle $ x $ at circumference.

Possibly, $ x $ is an inscribed angle intercepting the difference or sum.

But likely, $ x $ is half the difference of two arcs, if it's an angle formed by two chords.

For example, if $ x $ is formed by two chords intersecting, and the intercepted arcs are $ 120^\circ $ and $ 101^\circ $, then:

$ x = \frac{1}{2}(120^\circ - 101^\circ) = \frac{1}{2}(19^\circ) = 9.5^\circ $

Or if it's a tangent and chord, etc.

But without diagram, hard.

---

Final Answer Attempt:

Given the ambiguity, here’s what we can definitively say:

#### From the first part:

- $ \angle CPX = 28^\circ $ → central angle → $ \overarc{CX} = 28^\circ $
- $ \angle CPY = 38^\circ $ → $ \overarc{CY} = 38^\circ $
- $ \overarc{XY} = |\overarc{CY} - \overarc{CX}| = |38^\circ - 28^\circ| = 10^\circ $ (assuming same side)
- $ \angle XPY = 10^\circ $
- $ \overarc{WZ} = 180^\circ $ (diameter)

So:

1. $ \overarc{PZ} $: Invalid — P is center, not on circle
2. $ \overarc{WK} $: Unknown — no info about K
3. $ \angle NPZ $: No info about N
4. $ \overarc{VWZ} $: Arc from V to Z via W — depends on configuration
5. $ \angle XPY = 10^\circ $
6. $ \overarc{XY} = 10^\circ $
7. $ \overarc{WZ} = 180^\circ $

For problems 8–20, since I can’t see diagrams, I can’t give exact answers.

But if you can describe one of the diagrams (e.g., “Problem 17 has a triangle with two radii and an angle x at the center”), I can help.

---

Recommendation:

Please describe one or two of the diagrams (e.g., “Problem 17: circle with center O, triangle OPQ, with OP and OQ radii, angle at P is 30°, find angle at O”) and I’ll solve it.

Otherwise, the most I can do is provide general rules.

---

Summary of Rules:

- Central angle = intercepted arc
- Inscribed angle = half the intercepted arc
- Angle formed by two chords = half the sum of intercepted arcs
- Angle formed by tangent and chord = half the intercepted arc
- Diameter forms 180° arc
- Right angle inscribed in semicircle

Let me know if you can describe the diagrams or provide more details!