Circle Theorems (A) - A geometry worksheet with 12 problems to find missing angles using circle theorems.

Let’s solve each problem step-by-step using circle theorems. The key theorems we’ll use include:

- Angle at the center is twice the angle at the circumference (subtended by same arc).
- Angles in the same segment are equal.
- Angle in a semicircle is 90°.
- Opposite angles in a cyclic quadrilateral sum to 180°.
- Tangent-radius theorem: angle between tangent and radius is 90°.
- Alternate segment theorem: angle between tangent and chord equals angle in alternate segment.

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🔹 Problem 1:

We have an isosceles triangle inside a circle with apex angle = 116°, and two base angles labeled a.

Since it's isosceles (two sides are radii → equal), the base angles are equal.

Sum of angles in triangle = 180°

So:
a + a + 116° = 180°
→ 2a = 64°
→ a = 32°

✔ Answer: a = 32°

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🔹 Problem 2:

Triangle with angles 72°, 47°, and unknown b, c.

Wait — this looks like a triangle inscribed in a circle. But angles given are 72° and 47°, so third angle b = 180 - 72 - 47 = 61°

But wait — there’s also an angle c marked on the diagram. Looking closely, c is the angle at the center subtended by the same arc as angle b.

By angle at center = 2 × angle at circumference:

→ c = 2 × b = 2 × 61° = 122°

✔ b = 61°, c = 122°

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🔹 Problem 3:

We have a triangle with angles 54°, 67°, and unknown d, e, f.

First, find angle d — it’s part of a triangle with 54° and 67°.

So: d = 180 - 54 - 67 = 59°

Now, e is the angle at the center subtended by the same arc as d → e = 2 × d = 118°

Then f is another angle in the triangle — wait, actually, looking at the diagram, f is likely the angle at the center for the other arc? Wait — perhaps f is the reflex angle?

Actually, the central angle corresponding to arc opposite 54° would be 2×54=108°, but that’s not labeled.

Wait — re-examining: d is at circumference, e is central angle above it → e = 2d = 118°

Then f is probably the remaining angle in the triangle formed by two radii and chord — but since we already used all three angles, perhaps f is mislabeled? Or maybe f is the angle at the center for the 67° angle?

Actually, let’s assume:

- Angle subtended at circumference = 54° → central angle = 108°
- Angle subtended at circumference = 67° → central angle = 134°
- But we already have d = 59° (third angle)

Wait — perhaps f is the angle at the center for the 59° angle? That would be 118°, which is e.

This suggests a labeling issue. Let me reinterpret.

Looking at standard layout:

In problem 3, we have a triangle with angles 54°, 67°, and d (at circumference). So d = 59°

Then e is the central angle for arc opposite d → e = 118°

Then f is likely the central angle for the arc opposite the 54° angle → f = 2 × 54° = 108°

✔ So: d = 59°, e = 118°, f = 108°

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🔹 Problem 4:

We have a tangent and a chord. Angle between tangent and chord is 52°. We need to find g.

By alternate segment theorem: angle between tangent and chord = angle in alternate segment.

So g = 52°

✔ g = 52°

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🔹 Problem 5:

Triangle with angles 25°, 14°, and unknown h, i

First, h is the third angle in the triangle:
h = 180 - 25 - 14 = 141°

Then i is the angle at the center subtended by the same arc as h → i = 2 × h = 282°? That can’t be right — angles in circle can’t exceed 360°, but central angle can be reflex.

Wait — if h is 141°, then the central angle is 2×141 = 282°, which is a reflex angle. Possible.

But let’s check: Is h really 141°? That seems large for a triangle inscribed in a circle.

Wait — perhaps h is not the angle in the triangle? Looking again: the triangle has angles 25° and 14°, so yes, third angle is 141°. That’s valid.

So i = 2 × 141° = 282°

✔ h = 141°, i = 282°

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🔹 Problem 6:

We have a triangle with one angle 25°, and we need to find j.

It appears to be an isosceles triangle with two radii → base angles equal.

The apex angle is 25°, so base angles = (180 - 25)/2 = 77.5°

But j is labeled at the base — so j = 77.5°

✔ j = 77.5°

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🔹 Problem 7:

Triangle with angles 81°, 60°, and unknown k, l, m

First, k = 180 - 81 - 60 = 39°

Then l is central angle for arc opposite k → l = 2 × 39° = 78°

Then m is central angle for arc opposite 81° → m = 2 × 81° = 162°

✔ k = 39°, l = 78°, m = 162°

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🔹 Problem 8:

We have a cyclic quadrilateral with one angle 142°, and another 25°. Need to find n.

In a cyclic quadrilateral, opposite angles sum to 180°.

So if 142° is one angle, its opposite is n = 180 - 142 = 38°

✔ n = 38°

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🔹 Problem 9:

We have a circle with central angle 252°, and we need to find p.

The angle p is at the circumference, subtended by the same arc as the central angle.

But note: the central angle given is 252° — that’s the reflex angle. The minor arc is 360 - 252 = 108°.

The angle at the circumference is half the arc it subtends.

If p is subtended by the minor arc, then p = 108° / 2 = 54°

If it’s subtended by the major arc, it would be 252°/2 = 126°, but typically we take the smaller one unless specified.

Looking at the diagram, p is likely the angle in the segment — so p = 54°

✔ p = 54°

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🔹 Problem 10:

Triangle with angles 76°, and we need to find q.

Looks like an isosceles triangle with two radii → base angles equal.

Apex angle = 76°, so base angles = (180 - 76)/2 = 52°

q is a base angle → q = 52°

✔ q = 52°

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🔹 Problem 11:

Central angle = 122°, need to find r.

Angle r is at the circumference, subtended by the same arc → r = 122° / 2 = 61°

✔ r = 61°

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🔹 Problem 12:

We have a triangle with angles 35°, 40°, and unknown s, t

First, s = 180 - 35 - 40 = 105°

Then t is the angle at the center subtended by the same arc as s → t = 2 × 105° = 210°

✔ s = 105°, t = 210°

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## ✔ Final Answers:

1) a = 32°
2) b = 61°, c = 122°
3) d = 59°, e = 118°, f = 108°
4) g = 52°
5) h = 141°, i = 282°
6) j = 77.5°
7) k = 39°, l = 78°, m = 162°
8) n = 38°
9) p = 54°
10) q = 52°
11) r = 61°
12) s = 105°, t = 210°

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📌 *Note: Some answers involve reflex angles (like 282°, 210°), which are valid in circle geometry when referring to the larger arc.*