Educational worksheet on central and inscribed angles with diagrams and practice problems.

Let’s solve each example step by step.

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Example 1: Find the missing arc.

We see a circle with a central angle of 160°. The whole circle is 360°.
The missing arc (labeled x°) is the rest of the circle.

So:
x = 360 - 160
x = 200

✔ Missing arc = 200°

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Example 2: Find the missing arc.

This one shows an inscribed angle? Wait — actually, looking at the diagram: there’s a point on the circle, and two lines going out — one is a chord, one is a tangent? But it says “100°” next to the angle formed between the chord and the tangent. That’s a tangent-chord angle.

Rule: A tangent-chord angle equals half the measure of the intercepted arc.

So if the angle is 100°, then the intercepted arc is 2 × 100° = 200°.

But wait — that can’t be right because the angle shown is *outside* the circle? Let me recheck.

Actually, in the diagram for Example 2:
There’s a circle. One line is a diameter (horizontal). Another line goes from the endpoint of the diameter downward as a tangent. The angle between the diameter and the tangent is labeled 100°? That doesn’t make sense geometrically — a tangent at the end of a diameter should form a 90° angle with the radius/diameter.

Wait — perhaps the 100° is the inscribed angle? No, the vertex is on the circle, but one side is a tangent.

Actually, standard rule:
> The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

In this case, the angle is 100°, so the intercepted arc would be 200° — but that’s more than half the circle. And the missing arc x° is the other part.

If the intercepted arc is 200°, then the rest of the circle is 360 - 200 = 160°.

But let’s think again — maybe the 100° is NOT the tangent-chord angle. Looking at the diagram description: it says “100°” near the angle at the point where the tangent meets the circle, and the arc opposite is labeled x°.

Actually, another possibility: maybe the 100° is the central angle? No, the vertex is on the circle.

Wait — perhaps I misread. Let me reinterpret based on common problems.

Alternative approach: In many textbooks, when you have a tangent and a secant or chord forming an angle outside, but here the vertex is ON the circle.

Standard theorem:
Angle formed by tangent and chord = ½ × intercepted arc

So if angle = 100°, then intercepted arc = 200°. Then the remaining arc (the one not intercepted) is 360 - 200 = 160°.

But the problem says “Find the missing arc” and labels x° on the bottom arc — which is likely the non-intercepted one.

So x = 160°

BUT — 100° is too big for a tangent-chord angle unless the arc is major. It’s possible.

Alternatively, maybe the 100° is the reflex angle? Unlikely.

Wait — let’s look at Example 4 later for context. Maybe I’m overcomplicating.

Another thought: Perhaps the 100° is the inscribed angle subtending arc x°? But no — the vertex is on the circle, and one side is tangent.

I recall: If an angle is formed by a tangent and a chord, its measure is half the difference of the arcs? No — that’s for two secants or secant-tangent from outside.

For tangent and chord with vertex on circle: it’s simply half the intercepted arc.

So if angle = 100°, intercepted arc = 200°, so the other arc (x°) = 360 - 200 = 160°.

Yes, that must be it.

✔ Missing arc = 160°

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Example 3: Find the missing angle.

We have a circle with two inscribed angles sharing the same arc? Let’s see:

One inscribed angle is 68°, intercepting some arc. Another inscribed angle is x°, and they both seem to intercept the same arc? Or different?

Looking at the diagram description: There’s an arc labeled 136°. An inscribed angle of 68° is drawn, and another inscribed angle x° is also drawn, probably intercepting the same arc or adjacent.

Recall: Inscribed angle = ½ × intercepted arc.

So if an arc is 136°, then any inscribed angle intercepting that arc should be 136 / 2 = 68° — which matches the given 68°.

Now, what about x°? It looks like x° is another inscribed angle, but intercepting a different arc.

Wait — perhaps x° and 68° are angles in a triangle inside the circle? Or maybe they share a side.

Another idea: Maybe the 136° arc is intercepted by the 68° angle, and x° is intercepting the rest of the circle? But that wouldn’t make sense directly.

Perhaps the two angles are on the same chord? Like, they are angles subtended by the same chord but on opposite sides? Then they would be supplementary? Not necessarily.

Wait — let’s think differently. Suppose we have a cyclic quadrilateral? Not indicated.

Maybe x° is an inscribed angle intercepting the arc that is opposite to the 136° arc.

Total circle = 360°. Arc given = 136°. So the rest of the circle is 360 - 136 = 224°.

But x° is an inscribed angle — if it intercepts the 224° arc, then x = 224 / 2 = 112°.

Is that it? Let me check the diagram logic.

In many such problems, if you have one inscribed angle given and an arc, and another angle elsewhere, often the second angle intercepts the remaining arc.

Moreover, 68° intercepts 136° arc (since 68×2=136), so consistent.

Then x° intercepts the other arc: 360 - 136 = 224°, so x = 224 / 2 = 112°.

Yes, that makes sense.

✔ Missing angle = 112°

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Example 4: Find all arc and angles.

We have a circle with center marked. Two central angles are given: 52° and 64°. They are adjacent? Probably.

Since they are central angles, their measures equal the arcs they intercept.

So arc1 = 52°, arc2 = 64°.

What about the other arcs? The circle has 360° total.

Assuming these two angles are adjacent and together form part of the circle, the remaining arc would be 360 - 52 - 64 = 244°.

But are there only three arcs? The diagram might show four regions? Let me think.

Typically, if two radii are drawn creating two central angles, they divide the circle into two sectors? No — if you have two angles from the center, they might create three arcs if they are not opposite.

Wait — the diagram says "find all arc and angles", implying multiple.

Probably, the two given angles (52° and 64°) are adjacent central angles, so the arc between them is 52 + 64 = 116°? No — each central angle corresponds to its own arc.

Actually, each central angle defines an arc equal to itself.

So if there are two central angles shown, say angle AOB = 52°, angle BOC = 64°, then arc AB = 52°, arc BC = 64°, and arc CA (the rest) = 360 - 52 - 64 = 244°.

Now, are there any inscribed angles? The problem says "find all arc and angles" — probably meaning all central angles and arcs, and possibly inscribed angles if present.

But in the diagram description, it only mentions central angles 52° and 64°, and no other angles drawn. So likely, we just need to find the missing arc and confirm the central angles.

Perhaps there is an inscribed angle formed by connecting points? But not specified.

To be safe, let's assume we need to list:

- Central angles: 52°, 64°, and the third central angle for the remaining arc: 244°? But 244° is reflex, usually we take the smaller one.

Actually, central angles are typically taken as the smaller ones unless specified.

But 52 + 64 = 116, so the minor arc between the outer points is 116°, and the major arc is 244°.

But the central angle corresponding to the major arc would be 244°, which is unusual but mathematically correct.

Perhaps the diagram has only two central angles shown, and we need to find the arcs they intercept and the remaining arc.

Also, maybe there are inscribed angles formed by chords. For example, if we connect the endpoints, we might get inscribed angles.

Suppose points A, B, C on circle, center O. Angle AOB = 52°, angle BOC = 64°. Then arc AB = 52°, arc BC = 64°, arc AC (not passing through B) = 360 - 52 - 64 = 244°.

Now, if we draw chord AC, then an inscribed angle at B would intercept arc AC. So angle ABC = ½ × arc AC = ½ × 244° = 122°.

Similarly, angle at A intercepting arc BC: angle BAC = ½ × arc BC = ½ × 64° = 32°.

Angle at C intercepting arc AB: angle BCA = ½ × arc AB = ½ × 52° = 26°.

And indeed, in triangle ABC, angles sum to 32 + 26 + 122 = 180°, good.

So probably, the problem expects us to find all arcs and all relevant angles, including inscribed angles formed by the chords.

Given that, let's list:

Arcs:
- Arc AB = 52° (intercepted by central angle 52°)
- Arc BC = 64° (intercepted by central angle 64°)
- Arc AC (major) = 360 - 52 - 64 = 244°

Central angles:
- ∠AOB = 52°
- ∠BOC = 64°
- ∠AOC (reflex) = 244°, or the smaller one is 360 - 244 = 116°? Wait, no — if A to C via B is 52+64=116°, then the direct arc AC is 244°, so central angle for minor arc AC is min(244, 116)? Actually, the central angle is defined by the arc it subtends.

Typically, central angle is the smaller one, but in this case, since 52 and 64 are given separately, likely the figure has three points, and we have two small central angles, so the third central angle for the large arc is 244°, but we might report the minor arcs.

To avoid confusion, let's define:

Let the circle have center O, and points A, B, C on circumference, with ∠AOB = 52°, ∠BOC = 64°, and assuming B is between A and C along the minor arc.

Then:
- Minor arc AB = 52°
- Minor arc BC = 64°
- Minor arc AC = arc AB + arc BC = 52 + 64 = 116°? Only if B is on the minor arc AC.

Actually, if ∠AOB and ∠BOC are adjacent, then arc AC = arc AB + arc BC = 52 + 64 = 116°, and the reflex arc AC is 360 - 116 = 244°.

Central angles:
- ∠AOB = 52°
- ∠BOC = 64°
- ∠AOC = 116° (for minor arc AC)

Now, inscribed angles:
- Any inscribed angle intercepting arc AB: e.g., angle at C, ∠ACB = ½ × arc AB = 26°
- Intercepting arc BC: angle at A, BAC = ½ × arc BC = 32°
- Intercepting arc AC (minor): angle at B, ∠ABC = ½ × arc AC = ½ × 116° = 58°? But earlier I said 122° — mistake.

Ah, here's the key: if arc AC is 116° (minor), then inscribed angle at B intercepting arc AC is half of that, so 58°.

But in triangle ABC, angles at A, B, C should sum to 180°.

BAC = 32° (intercepts arc BC=64°)
∠BCA = 26° (intercepts arc AB=52°)
∠ABC = ? Should be 180 - 32 - 26 = 122°

Contradiction.

Why? Because if B is on the major arc, then the inscribed angle at B intercepting arc AC would be half the difference or something.

Standard rule: the measure of an inscribed angle is half the measure of its intercepted arc, and the intercepted arc is the one that the angle "sees", i.e., the arc not containing the angle's vertex.

So for triangle ABC inscribed in circle, angle at B intercepts arc AC that does not contain B.

If points are in order A, B, C around the circle, and arc AB=52°, arc BC=64°, then arc AC not containing B is the long way: 360 - 52 - 64 = 244°.

Therefore, angle at B intercepts arc AC of 244°, so ∠ABC = ½ × 244° = 122°.

Angle at A intercepts arc BC = 64°, so ∠BAC = 32°.

Angle at C intercepts arc AB = 52°, so ∠BCA = 26°.

Sum: 32 + 26 + 122 = 180°, perfect.

So arcs:
- Arc AB = 52°
- Arc BC = 64°
- Arc AC (major, not containing B) = 244°

Central angles:
- ∠AOB = 52°
- ∠BOC = 64°
- AOC (reflex) = 244°, or the smaller central angle for arc AC is min(244, 116) = 116°, but since the arc not containing B is 244°, the central angle for that arc is 244°.

Usually, we report the minor arcs unless specified.

But for completeness, let's list all.

The problem says "find all arc and angles", so likely:

Arcs:
- Minor arc AB: 52°
- Minor arc BC: 64°
- Major arc AC: 244° (or minor arc AC: 116° if we go the short way)

But in the context, since the central angles given are 52° and 64°, and they are adjacent, the arc from A to C via B is 116°, so minor arc AC is 116°, and major is 244°.

Inscribed angles:
- ∠ABC = 122° (intercepts major arc AC)
- ∠BAC = 32° (intercepts arc BC)
- ∠BCA = 26° (intercepts arc AB)

Also, the central angles are given as 52° and 64°, and the third central angle for minor arc AC is 116°.

So to summarize for Example 4:

Arcs:
- Arc AB = 52°
- Arc BC = 64°
- Arc AC (minor) = 116°
- Arc AC (major) = 244°

Angles:
- Central: ∠AOB = 52°, ∠BOC = 64°, ∠AOC = 116° (for minor arc)
- Inscribed: ∠BAC = 32°, ∠BCA = 26°, ∠ABC = 122°

But perhaps the problem only wants the arcs corresponding to the given central angles and the remaining arc, and maybe the inscribed angles if implied.

Since the instruction is "find all arc and angles", and given the diagram likely shows the two central angles and the chords, I'll provide the main ones.

To keep it simple and accurate:

From the given:
- Central angle 52° → arc = 52°
- Central angle 64° → arc = 64°
- Remaining arc = 360 - 52 - 64 = 244°

Inscribed angles formed by the chords:
- Angle intercepting 52° arc: 26°
- Angle intercepting 64° arc: 32°
- Angle intercepting 244° arc: 122°

So for the answer, I'll list them clearly.

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Now, compiling final answers for all examples:

Example 1:
Missing arc = 360 - 160 = 200° → 200

Example 2:
Tangent-chord angle = 100° → intercepted arc = 200° → missing arc = 360 - 200 = 160° → 160

Example 3:
Given arc = 136°, inscribed angle = 68° (which is half, correct). Missing angle x° intercepts the remaining arc: 360 - 136 = 224° → x = 224 / 2 = 112° → 112

Example 4:
Arcs: 52°, 64°, and 244° (since 360 - 52 - 64 = 244)
Inscribed angles: half of each arc: 26°, 32°, 122°
Central angles: 52°, 64°, and 244° (or 116° for minor, but 244° is the actual for the major arc)

But to match typical expectations, perhaps list:

- Arcs: 52°, 64°, 244°
- Angles: central 52°, 64°, 244°; inscribed 26°, 32°, 122°

Since the problem says "find all arc and angles", and given the diagram, I'll provide the numerical values.

For clarity in final answer, I'll box each example's answer.

Final Answer:

Example 1: 200
Example 2: 160
Example 3: 112
Example 4: Arcs: 52°, 64°, 244°; Angles: central 52°, 64°, 244°; inscribed 26°, 32°, 122°

But to format neatly:

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Final Answer:
Example 1: 200
Example 2: 160
Example 3: 112
Example 4: Arcs: 52°, 64°, 244°; Central angles: 52°, 64°, 244°; Inscribed angles: 26°, 32°, 122°