Arcs, Central Angles & Inscribed Angles Worksheet - Free Printable

Let’s solve each problem step by step. We’ll use circle geometry rules:

- Central angle = measure of its intercepted arc.
- Inscribed angle = half the measure of its intercepted arc.
- If two chords intersect inside a circle, the angle formed is half the sum of the intercepted arcs.
- If an angle is formed by a tangent and a chord, it’s half the intercepted arc.
- Diameter → 180° arc.
- Full circle → 360°.

We’re told:
∠WPX = 28°, ∠ZPY = 38°, WZ and XV are diameters.

Also, O is center in problems 5–20.

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Problem 1: VZ

VZ is an arc. Since XV is a diameter, arc XV = 180°.
Point P is on the circle? Wait — looking at diagram (we can’t see but from context), points W, X, Y, Z, V are on circle, P is intersection point of chords? Actually, since WZ and XV are diameters, they cross at center O? But problem says “in ⊙P” — wait, that must be typo? Probably ⊙O? Or maybe P is center? Let me re-read.

Wait: “Find each measure in ⊙P if m∠WPX = 28°, m∠ZPY = 38°, and WZ and XV are diameters.”

If it’s ⊙P, then P is the center! That makes sense. So P is center. Then ∠WPX and ∠ZPY are central angles.

So:

- ∠WPX = 28° → arc WX = 28°
- ∠ZPY = 38° → arc ZY = 38°
- WZ is diameter → arc WZ = 180°
- XV is diameter → arc XV = 180°

Now, let’s label points around circle. Assume order: W, X, Y, Z going around? Not sure. But since WZ and XV are diameters, they cross at center P.

So lines W-P-Z and X-P-V are straight lines (diameters).

Then angle between them: ∠WPX = 28°, so angle between diameter WZ and XV is 28° at center.

Similarly, ∠ZPY = 38° — Z to P to Y. Since Z-P-W is straight, and X-P-V is straight, we can find other angles.

Actually, since WZ and XV are diameters intersecting at P (center), vertical angles are equal.

∠WPX = 28° → then vertically opposite angle ∠ZPV = 28°

∠ZPY = 38° → then vertically opposite angle ∠XPW? Wait no.

Let me draw mentally:

Lines: W—P—Z (horizontal?), X—P—V (diagonal?)

Angle between WP and XP is 28° → so arc WX = 28°

Angle between ZP and YP is 38° → but where is Y?

Probably Y is on the circle such that PY is a radius? But not necessarily diameter.

Wait — perhaps all points W,X,Y,Z,V are on circle, P is center, WZ and XV are diameters, so W,P,Z colinear; X,P,V colinear.

Then angles given are central angles.

So:

- ∠WPX = 28° → arc WX = 28°
- ∠ZPY = 38° → arc ZY = 38°

Now, since WZ is diameter, arc WZ = 180°. Arc WZ goes from W to Z passing through... which way? The minor or major? Usually minor unless specified.

But with diameters crossing, the circle is divided into 4 arcs: WX, XY, YZ, ZW? Not exactly.

Actually, the two diameters divide the circle into 4 arcs.

Let’s define the arcs between the endpoints.

The four points: W, X, Z, V? But Y is also there.

Perhaps the points are arranged as W, X, Y, Z around the circle, with diameters WZ and XV.

Since XV is diameter, V is opposite X. Similarly, Z is opposite W.

So positions:

Assume starting from W, then moving to X, then to Y, then to Z, then back to W.

Arc WX = ? Given ∠WPX = 28°, and P is center, so yes, arc WX = 28°.

Now, from X to V is 180° because XV is diameter. So arc XV = 180°.

Arc XV includes arc XY + arc YV? But we don't know Y yet.

∠ZPY = 38°. Z to P to Y. Since Z is opposite W, and P is center, vector PZ is opposite PW.

Angle between PZ and PY is 38°, so arc ZY = 38°.

Now, total circle 360°.

Diameters WZ and XV intersect at P, so they form vertical angles.

Angle between WP and XP is 28°, so angle between ZP and VP should also be 28° (vertical angles).

Angle between ZP and YP is 38°, so Y is somewhere.

Let me assign directions.

Set point W at 0°, so Z at 180° (since WZ diameter).

XV is another diameter. Angle between WP and XP is 28°. Since WP is along 0°, XP could be at 28° or -28°. Let's say XP is at 28°, so V is at 28° + 180° = 208°.

Now, ∠ZPY = 38°. Z is at 180°, P is center, so PY is at some angle θ, and angle between 180° and θ is 38°, so θ = 180° ± 38° = 218° or 142°.

Now, we need to see which one makes sense. Probably Y is between X and Z or something.

Arcs:

From W (0°) to X (28°): arc WX = 28°

From X (28°) to V (208°): but that's 180°, good.

From V (208°) to Z (180°)? 208 to 180 is backwards, better to go increasing.

List points in order around circle.

Assume angles from positive x-axis:

- W: 0°

- X: 28° (since ∠WPX=28°)

- V: 28° + 180° = 208° (opposite X)

- Z: 180° (opposite W)

Now Y: ∠ZPY=38°, Z is at 180°, so PY is at 180° + 38° = 218° or 180° - 38° = 142°.

If Y is at 218°, then it's after V (208°), before Z? 218 > 208, and Z is 180, so from 208 to 218 to 360/0 to 180? Messy.

If Y is at 142°, then between X (28°) and Z (180°).

28° to 142° to 180°.

That might work.

Check arc ZY: from Z 180° to Y 142°? That would be |180-142|=38°, but direction: if going clockwise or counterclockwise.

Arc measure is usually the smaller one, but in context, since angle at center is 38°, arc ZY should be 38°, so the minor arc between Z and Y is 38°.

If Z at 180°, Y at 142°, difference 38°, yes, minor arc is 38°.

Similarly, if Y were at 218°, difference |218-180|=38°, also minor arc 38°.

Now, which one? We have point V at 208°, so if Y at 218°, it's close to V.

But let's see the diagrams later; for now, both possible, but probably Y is not near V.

In problem 3, it asks for ∠VPZ, which might help.

Perhaps from the names, the order is W,X,Y,Z around the circle.

So assume W at 0°, X at 28°, Y at ? , Z at 180°.

Then arc WX = 28°, arc XY = ?, arc YZ = ?, arc ZW = 180°? No, arc from Z to W is 180° only if direct, but with points in between.

Total from W to Z via X and Y should be less than 180 if minor, but WZ is diameter, so arc WZ is 180° whichever way, but the minor arc is 180° since diameter.

I think I'm confusing myself.

Let me define the arcs between the diameter endpoints.

The two diameters WZ and XV intersect at P, forming four central angles.

Given ∠WPX = 28°, which is the angle at P between points W,P,X.

Since WZ and XV are straight lines, the vertical angle to ∠WPX is ∠ZPV, so ∠ZPV = 28°.

Also, adjacent angles sum to 180°.

So, angle between WP and ZP is 180° (straight line), so angle between XP and ZP is 180° - 28° = 152°.

Similarly, angle between WP and VP is 180° - 28° = 152°, etc.

Now, we're given ∠ZPY = 38°. This is the angle at P between Z,P,Y.

Y is another point on the circle, so PY is a radius.

So, from ray PZ, we move 38° to ray PY.

Depending on direction, PY could be 38° from PZ towards PW or towards PV.

In the circle, likely Y is in the arc not containing W and X or something.

To resolve, let's look at problem 3: ∠VPZ.

∠VPZ is the angle at P between V,P,Z.

From above, since WZ and XV are diameters, and ∠WPX = 28°, then the angle between V and Z.

Ray PV is opposite to PX, ray PZ is opposite to PW.

So angle between PV and PZ: since angle between PW and PX is 28°, and PV is 180° from PX, PZ is 180° from PW, so the angle between PV and PZ should be the same as between PW and PX, which is 28°, because it's vertical angles.

Let's think vectors.

Suppose ray PW is 0°, then ray PZ is 180°.

Ray PX is at 28° (since ∠WPX=28°), so ray PV is at 28° + 180° = 208°.

Then angle between PV (208°) and PZ (180°) is |208 - 180| = 28°, yes.

So ∠VPZ = 28°.

But that's problem 3, and we have ∠ZPY = 38°, which is separate.

For ∠ZPY = 38°, from ray PZ (180°), ray PY is at 180° + 38° = 218° or 180° - 38° = 142°.

Now, if we take Y at 142°, then it's between X (28°) and Z (180°).

If at 218°, between V (208°) and W (0°=360°).

In many diagrams, Y is likely in the upper part, so perhaps 142°.

Moreover, in problem 4, it asks for VW, which might be arc or chord, but probably arc.

Let's calculate the arcs.

Assume Y is at 142°.

So points:

- W: 0°

- X: 28°

- Y: 142°

- Z: 180°

- V: 208°

Now, check arc ZY: from Z 180° to Y 142°. The minor arc is min(|180-142|, 360-|180-142|) = min(38, 322) = 38°, good.

Arc WX: 28° to 0° = 28°, good.

Now, problem 1: VZ

V at 208°, Z at 180°, so arc VZ = |208 - 180| = 28°, but is this the minor arc? 28° < 180°, yes.

But let's confirm the path. From V 208° to Z 180°, if going clockwise, 208 to 360/0 to 180 is 152° + 180° = 332°, too big. Counterclockwise from 208 to 180 is 28°, yes, minor arc is 28°.

But is this correct? In the circle, with points at 0,28,142,180,208, the arc from V to Z directly is 28°, but there might be other points.

Perhaps VZ means the arc not containing other points, but usually it's the minor arc unless specified.

But let's see problem 2: WX — that's given as 28°, since central angle 28°.

Problem 3: ∠VPZ — we said 28°.

Problem 4: VW — V at 208°, W at 0°=360°, so arc VW = |360 - 208| = 152°, or the other way 208°, so minor is 152°.

But let's list all.

First, for problem 1: VZ

With Y at 142°, V at 208°, Z at 180°, so arc VZ = 208 - 180 = 28°.

But is there a reason to choose Y at 218°? Let's try that.

If Y at 218°, then arc ZY = |218 - 180| = 38°, good.

Points: W0°, X28°, Z180°, V208°, Y218°.

Then arc VZ = |208 - 180| = 28° same.

Arc VW = from V208° to W0°=360°, so 360-208=152°.

Same as before.

Now, what about arc XY or something.

In this case, with Y at 218°, then from X28° to Y218°, arc is 190°, which is large, while if Y at 142°, arc XY = 142-28=114°, more reasonable.

Also, in problem 5, it has ∠XPY, which would be angle at P between X,P,Y.

If Y at 142°, X at 28°, so angle |142-28| = 114°.

If Y at 218°, |218-28| = 190°, which is reflex, usually we take smaller angle, so 170°? 360-190=170°, still large.

Whereas 114° is nicer.

Moreover, in the diagram descriptions, likely Y is between X and Z.

So I'll assume Y at 142°.

So summary:

- W: 0°

- X: 28°

- Y: 142°

- Z: 180°

- V: 208°

Now, arcs:

1. VZ: from V 208° to Z 180°. Minor arc is 28° (since 208-180=28).

2. WX: from W 0° to X 28° = 28°.

3. ∠VPZ: angle at P between V,P,Z. Rays at 208° and 180°, difference 28°, so 28°.

4. VW: from V 208° to W 0°=360°, so 360-208=152°.

5. ∠XPY: X at 28°, Y at 142°, so angle |142-28| = 114°.

6. XY: arc from X 28° to Y 142° = 114°.

7. XWY: this might be arc X to W to Y, but usually it's the arc from X to Y passing through W.

From X 28° to Y 142° via W: X to W is 28° (but backwards), better to calculate the long way.

Arc XWY: from X to Y via W. So from X 28° to W 0°=360°, then to Y 142°? That doesn't make sense.

Typically, arc XWY means the arc from X to Y that passes through W.

So from X 28° to W 0° (which is 360°), then to Y 142°? But 142° is before 360 in numerical, but in circle, from 28° to 360° is 332°, then to 142° is additional, but that's not standard.

The measure of arc XWY is the sum of arc XW and arc WY.

Arc XW: from X to W. Since W is at 0°, X at 28°, minor arc is 28°, but if going from X to W via the short way, it's 28°, but for arc XWY, it might be the major arc.

Standard notation: arc ABC means from A to C passing through B.

So arc XWY: from X to Y passing through W.

So path: X -> W -> Y.

From X 28° to W 0°: if we go decreasing angle, 28° to 0° is 28°, but in terms of arc measure, it's the length, so 28°.

Then from W 0° to Y 142°: 142° - 0° = 142°.

So total arc XWY = 28° + 142° = 170°.

Is that correct? From X to W is 28° (minor), W to Y is 142°, so yes, total 170°.

The other way from X to Y directly is 114°, so this is the major arc.

8. WZX: arc from W to X passing through Z.

W to Z to X.

W 0° to Z 180°: 180°.

Z 180° to X 28°: from 180 to 28, which is 180 to 360/0 to 28, so 180° + 28° = 208°? No.

From Z 180° to X 28°, the minor arc is min(|180-28|, 360-152) = min(152, 208) = 152°, but since we are going through the long way for arc WZX, it should be the arc not containing Y and V.

From W to Z is 180° (diameter), then Z to X: if we go from Z 180° to X 28°, the short way is clockwise 152° (180 to 360 is 180, minus 28? From 180 to 28 clockwise: 180 to 360 is 180°, then 0 to 28 is 28°, total 208°, but that's not right.

Angles: from 180° to 28°, the difference is 152° if going counterclockwise (180 down to 28 is 152°? 180 - 28 = 152°, yes, since 28 < 180, counterclockwise from 180 to 28 is 152°.

Clockwise from 180 to 28 is 360 - 152 = 208°.

For arc WZX, from W to X via Z, so W to Z is 180° (say counterclockwise from 0 to 180), then Z to X: if we continue counterclockwise from 180 to 28, but 28 is less, so we go to 360 then to 28, which is 180° + 28° = 208°? From 180 to 360 is 180°, 0 to 28 is 28°, total 208°.

Since the whole circle is 360°, and arc WX minor is 28°, so arc WZX should be 360° - 28° = 332°? That can't be.

I think I have a mistake.

Arc WZX: starts at W, ends at X, passes through Z.

So the arc is W -> Z -> X.

The measure is the sum of arc WZ and arc ZX.

Arc WZ: since WZ is diameter, arc WZ = 180° (minor arc, but actually for diameter, both arcs are 180°, so we can take 180°).

Arc ZX: from Z to X. Points Z 180°, X 28°. The minor arc is 152° (as |180-28|=152), but when we go from Z to X in the path W-Z-X, if we are going the long way, it should be the major arc from Z to X.

From Z 180° to X 28°, the arc not containing W. Since W is at 0°, and Z at 180°, X at 28°, so from Z to X, the arc passing through V and Y would be from 180 to 208 to 218? Earlier I had Y at 142, V at 208.

From Z 180° to X 28°, the short way is counterclockwise 152° (180 to 28 is 152° decrease), passing through Y at 142°.

The long way is clockwise from 180 to 360/0 to 28, which is 180° + 28° = 208°, passing through V at 208°? V is at 208, which is on the way.

From 180 to 208 is 28°, then 208 to 360 is 152°, then 0 to 28 is 28°, total 28+152+28=208°, yes.

So arc ZX long way is 208°.

Then arc WZ is 180°, but if we add arc WZ and arc ZX, we double-count or something.

When we say arc WZX, it is the arc from W to X via Z, so the total measure is the length from W to Z plus Z to X along that path.

From W 0° to Z 180°: if we go counterclockwise, 180°.

Then from Z 180° to X 28° along the long way: as above, 208°? But 180° + 208° = 388° > 360, impossible.

The issue is that from W to Z is 180°, and from Z to X via the long way is the arc not containing W, which is 208°, but when combined, from W to X via Z long way should be the entire circle minus the minor arc WX.

Minor arc WX is 28°, so major arc W to X is 360° - 28° = 332°.

And this major arc passes through Z, since Z is opposite, so yes, arc WZX = 332°.

Similarly, for arc XWY, from X to Y via W, minor arc XY is 114°, so major arc is 360° - 114° = 246°, but earlier I calculated 28° + 142° = 170°, which is wrong because from X to W is 28°, but from W to Y is 142°, but 28+142=170, and 360-170=190, not matching.

Let's calculate the actual arc measures.

From X 28° to Y 142°: difference 114°, so minor arc XY = 114°.

Major arc XY = 360° - 114° = 246°.

This major arc passes through W and Z and V.

Specifically, from X 28° to W 0°=360°: 332°? From 28 to 360 is 332°, but that's not to Y.

From X 28° to Y 142° via W: so from 28° to 0°=360° (arc of 332°? No, from 28 to 360 is 332°, but that's to W, then from W to Y 142° is 142°, but 332 + 142 = 474 > 360, nonsense.

The arc from X to Y via W means the arc that includes W, so it should be the arc from X to Y that contains W.

In the circle, with points at 0(W), 28(X), 142(Y), 180(Z), 208(V), the arc from X to Y containing W would be from X 28° to Y 142° going the long way around, passing through Z, V, W.

So from X 28° to Z 180°: 152° (counterclockwise), then Z 180° to V 208°: 28°, then V 208° to W 0°=360°: 152° (360-208=152), then W 0° to Y 142°: 142°, but that's not efficient.

From X 28° to Y 142° via the path that includes W: since W is at 0°, and 0° is between 28° and 142° if we go clockwise, but 28 to 0 is clockwise 28°, 0 to 142 is 142°, total 170°, and this arc does not include Z or V; it includes only from 28 down to 0 to 142, so points with angles from 0 to 28 and 0 to 142, so it includes W, but not Z or V.

And the measure is 28° (X to W) + 142° (W to Y) = 170°.

The other arc from X to Y is directly 114°, which does not contain W.

So for arc XWY, it should be 170°.

Similarly, for arc WZX: from W to X via Z. W at 0°, X at 28°, Z at 180°.

From W to Z: 180° (say to 180°), then Z to X: from 180° to 28°, which is 152° if going counterclockwise (180 to 28 is 152° down), but 180 + 152 = 332°, and this arc contains Y at 142°, since 142 is between 28 and 180.

From W 0° to Z 180° counterclockwise: 180°, passing through no other points yet.

Then from Z 180° to X 28° counterclockwise: from 180 to 28, which is decreasing angle, so 180 to 142 to 28, so 152°, and this includes Y at 142°.

So total arc W to X via Z is 180° + 152° = 332°, and it contains Y.

The minor arc WX is 28°, so major is 332°, yes.

So arc WZX = 332°.

Now back to problems.

1. VZ: V 208°, Z 180°, minor arc |208-180| = 28°.

2. WX: 28°.

3. ∠VPZ: angle at P between V,P,Z. Rays at 208° and 180°, difference 28°, so 28°.

4. VW: V 208°, W 0°=360°, minor arc min(360-208, 208) = min(152, 208) = 152°.

5. ∠XPY: X 28°, Y 142°, angle |142-28| = 114°.

6. XY: arc from X to Y minor = 114°.

7. XWY: arc from X to Y via W = as above, 28° (X to W) + 142° (W to Y) = 170°.

8. WZX: arc from W to X via Z = 180° (W to Z) + 152° (Z to X) = 332°, or 360° - 28° = 332°.

Now, problems 5-20 have O as center, and different diagrams.

But for now, let's finish 1-8.

So:

1. VZ = 28°

2. WX = 28°

3. ∠VPZ = 28°

4. VW = 152°

5. ∠XPY = 114°

6. XY = 114°

7. XWY = 170°

8. WZX = 332°

Now, for problems 5-20, O is center, and we need to find x or justify.

But since no diagrams, I'll assume standard configurations.

Problem 5: triangle with angles 4x, 2x, and at center? It says "calculate the value of x", and diagram has a triangle inscribed or something.

Typically, if it's a triangle with vertices on circle, and O center, but here it might be a triangle with one vertex at center.

Looking at description: "5. [diagram] 4x, 2x" — probably a triangle with angles 4x and 2x at circumference, and we need to find x.

But without diagram, hard.

Perhaps it's a central angle and inscribed angle.

Common type: if there is a central angle and an inscribed angle subtending the same arc.

For example, in problem 5, likely there is an arc, and a central angle and an inscribed angle.

Suppose the diagram shows a circle with center O, and points A,B,C on circle, with angle at O being 4x, angle at B being 2x, and they subtend the same arc AC.

Then, central angle is twice the inscribed angle, so 4x = 2 * 2x, which is always true, not helpful.

Perhaps different arcs.

Another common type: triangle with vertices on circle, and O center, but O may not be vertex.

Perhaps it's a triangle formed by two radii and a chord.

For example, isosceles triangle with two sides radius, so base angles equal.

In problem 5: "4x, 2x" — likely the angles are given, and we need to find x.

Assume it's a triangle with angles 4x, 2x, and the third angle.

Sum of angles 180°.

But what is the third angle? If it's at center, or at circumference.

Perhaps the 4x is at center, 2x at circumference, but they may not be related directly.

Let's look at problem 6: "4x, 4x" — probably isosceles.

Problem 7: "4x, 2x" again.

Perhaps for problem 5: suppose there is a central angle of 4x, and an inscribed angle of 2x subtending the same arc, then 4x = 2 * 2x, identity.

Or perhaps the inscribed angle is subtending a different arc.

Another idea: in some diagrams, there is a triangle with one angle at center, and the other two at circumference, but that doesn't make sense.

Perhaps it's the angle at the center and the angle at the circumference for the same arc, but in a triangle.

Let's think of a specific example.

Suppose in problem 5, there is arc AB, central angle AOB = 4x, and inscribed angle ACB = 2x, but then 4x = 2*2x, so no new info.

Unless the 2x is not for the same arc.

Perhaps the triangle is OAB, with OA, OB radii, so isosceles, and angle at O is 4x, then base angles are (180-4x)/2 = 90-2x each.

But the diagram shows 4x and 2x, so perhaps one base angle is 2x, so 90-2x = 2x, then 90 = 4x, x=22.5.

That makes sense.

Similarly for others.

So for problem 5: likely triangle with vertex at center O, so OA and OB radii, angle at O is 4x, and one base angle is 2x.

Since isosceles, base angles equal, so both base angles are 2x.

Then sum: 4x + 2x + 2x = 8x = 180°, so x = 22.5°.

But the diagram might show only one base angle as 2x, implying the other is also 2x.

Yes.

Problem 6: "4x, 4x" — probably the two base angles are 4x each, so angle at O is 180 - 8x, but not given, or perhaps angle at O is given.

Diagram shows "4x, 4x", so likely the two equal angles are 4x, so if it's isosceles with apex at O, then base angles are 4x each, so angle at O is 180 - 8x.

But we need another equation. Perhaps the arc or something.

Maybe the 4x are the base angles, and we need to find x, but no other info, so probably the angle at O is involved.

Perhaps in some diagrams, the angle at circumference is given.

Another common type: inscribed angle and central angle.

For problem 6: if it's a triangle with two angles 4x, and it's isosceles, but where is O.

Perhaps O is the center, and the triangle is inscribed, with O not necessarily vertex.

But typically in such problems, for 5-8, it's triangles with O as a vertex.

Assume for problem 6: triangle OAB, OA=OB radii, so isosceles, and the two base angles are both 4x, so angle at O is 180 - 8x.

But we need to find x, so probably there is more, or perhaps the arc is given, but not.

Perhaps the "4x, 4x" are the angles at A and B, and we need to find x, but without relation, can't.

Unless the angle at O is implied to be known, but not.

Another possibility: in some diagrams, there is a central angle and an inscribed angle sharing the arc.

For example, in problem 6, perhaps there is arc AB, central angle AOB = y, inscribed angle ACB = 4x, and another angle.

But diagram shows "4x, 4x", so likely two angles are 4x.

Perhaps it's a quadrilateral or something.

Let's look at problem 7: "4x, 2x" again.

Problem 8: "4x, 2x" .

Perhaps for all, it's similar to problem 5.

For problem 6: if the two base angles are 4x, then angle at O is 180 - 8x, but if no other info, perhaps we can't find x, but that can't be.

Unless the angle at O is given as something else, but in the text, only "4x, 4x" is written, so probably the angles marked are 4x and 4x, and they are the base angles, so for isosceles triangle, sum is 180, so 4x + 4x + angle at O = 180, but angle at O is unknown.

Perhaps in the diagram, the angle at O is not marked, but we can infer.

Another idea: perhaps the "4x, 4x" are the angles at the circumference, and O is center, but for a triangle inscribed.

For example, triangle ABC inscribed, O center, but then angles at A,B,C are given, but not related directly to O.

Unless it's the central angles.

I recall that in some problems, for a triangle with vertices on circle, the angle at center is twice the angle at circumference for the same arc.

But for problem 6, if it's showing two angles of 4x, perhaps they are inscribed angles subtending the same arc, but then they should be equal, which they are, but no new info.

Perhaps it's a different configuration.

Let's consider problem 9: "x, 96°" — likely an inscribed angle and central angle or something.

Perhaps for problem 5: the diagram has a central angle of 4x, and an inscribed angle of 2x, but they are for different arcs, or perhaps the 2x is part of a triangle.

Another common type: the angle between tangent and chord, but not mentioned.

Perhaps in problem 5, there is a triangle with angles 4x at center, 2x at circumference, and the third angle is at circumference, and they are related by the arc.

Suppose arc AB has central angle 4x, then inscribed angle subtending the same arc is 2x, which matches, but then for the triangle, if it's triangle OAB, with O center, A,B on circle, then angle at O is 4x, angles at A and B are equal, say y each, so 4x + 2y = 180.

But the diagram shows 4x and 2x, so perhaps one of the base angles is 2x, so y = 2x, then 4x + 2*(2x) = 8x = 180, x=22.5, as before.

For problem 6: "4x, 4x" — if the two base angles are 4x each, then angle at O is 180 - 8x, but if no other info, perhaps the diagram implies that the angle at O is known, or perhaps it's not isosceles with apex at O.

Perhaps for problem 6, the 4x are the angles at A and B, and O is center, but the triangle is ABC, with O inside, but complicated.

Another possibility: in some diagrams, there is a central angle and the inscribed angle is half, but here both are given as 4x, so perhaps it's the same angle or something.

Let's look at problem 7: "4x, 2x" — same as 5.

Problem 8: "4x, 2x" .

Perhaps for 6, it's different.

Problem 6 might have the angle at O as 4x, and the base angles as 4x each, but then 4x + 4x + 4x = 12x = 180, x=15, but that would be equilateral, but 4x=60, ok, but why mark two as 4x if all are equal.

Usually, if all equal, they mark all or say equilateral.

Perhaps the "4x, 4x" are the base angles, and the angle at O is not marked, but we can find if we know the arc, but not.

Let's skip and do later ones.

Problem 9: "x, 96°" — likely an inscribed angle x and central angle 96° for the same arc, so x = 96° / 2 = 48°.

Or vice versa.

Usually, if central angle is 96°, inscribed angle is half, 48°.

So x = 4