Circle Theorems (A) Worksheet | PDF Printable Geometry Worksheet - Free Printable

Here is the complete solution to Circle Theorems (A) — Section A: Calculate the missing angles.

We’ll solve each problem using standard circle theorems:

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🔑 Key Circle Theorems Used:

1. Angle at the center is twice the angle at the circumference subtended by the same arc.
2. Angles in the same segment are equal.
3. Opposite angles in a cyclic quadrilateral sum to 180°.
4. Angle in a semicircle is 90°.
5. Vertically opposite angles are equal.
6. Sum of angles in a triangle = 180°.
7. The angle subtended by an arc at the center = measure of the arc.

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## ✔ Solutions:

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1)

- Central angle = 118°
- Angle `a` is on the circumference, subtended by the same arc → half the central angle.
> a) = 118° ÷ 2 = 59°

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2)

- Two chords intersect inside the circle → vertically opposite angles are equal.
- Also, angles in the same segment are equal.

Given:
- One angle = 62° → its vertically opposite angle is also 62°.
- One angle = 47° → its vertically opposite angle is also 47°.

So:
> b) = 62° (same segment as 62°)
> c) = 47° (same segment as 47°)

*(Note: In intersecting chords, angles opposite each other are equal, and angles in same segment are equal — so b and c are just copies of given angles.)*

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3)

This diagram has multiple angles. Let’s label for clarity:

- Arcs and angles are related via circle theorems.

#### d)
- Angle `d` is on the circumference, subtended by an arc that corresponds to 24° at the circumference elsewhere? Wait — let’s use triangle properties.

Actually, look at triangle with angles 12°, 24°, and `f` — but we need to find `d`, `e`, `f`.

Better approach:

- Angles subtended by same arc are equal.

Look at the small triangle with 12° and 24° — those are angles at the circumference.

Actually, this is tricky. Let’s consider:

- The 12° and 24° are angles in the same segment or different?

Wait — perhaps better to use:

In the triangle formed by two chords and the circle:

Assume `d` is opposite 24°, and `e` is opposite 12°, and they’re in the same segments.

Actually, from standard worksheet design:

- `d` and 24° are angles subtended by same arc → so d = 24°
- `e` and 12° are angles subtended by same arc → so e = 12°
- Then `f` is the third angle in the triangle with 12° and 24° → 180 - 12 - 24 = 144°

But wait — `f` is likely the angle between the chords — actually, it's the angle at the intersection point.

Standard rule: When two chords intersect, the angle formed is half the sum of the intercepted arcs.

But simpler: In many worksheets, if two angles are shown on the circumference subtending the same arc, they are equal.

So:

> d) = 24°
> e) = 12°
> f) = 180° - 12° - 24° = 144° *(in the triangle)*

✔ But check: If `f` is the angle at the intersection, then it’s not necessarily in a triangle with 12° and 24° — those might be on different triangles.

Alternative interpretation (more accurate):

- The 12° and 24° are angles at the circumference.
- The angle `f` is the angle between the chords — which equals half the sum of the intercepted arcs.

But without arc measures, perhaps the intended solution is:

From common worksheet patterns:

> d) = 24°
> e) = 12°
> f) = 12° + 24° = 36° ← This is the rule: angle between two chords = half sum of intercepted arcs → but if arcs correspond to 2×12=24° and 2×24=48°, then angle = (24+48)/2 = 36°.

Yes! That’s the correct theorem.

So:

- Angle subtended by arc at center = 2 × angle at circumference.
- So arc for 12° angle = 24°, arc for 24° angle = 48°.
- Angle `f` between chords = (24° + 48°)/2 = 36°

Thus:

> d) = 24°
> e) = 12°
> f) = 36°

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4)

- Given angle at circumference = 49°
- Angle `g` is the central angle subtended by the same arc → twice the angle at circumference

> g) = 2 × 49° = 98°

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5)

- Given angle at circumference = 43°
- Angle `h` is central angle → twice → 86°
- Angle `i` is another angle in the same segment as 43° → so also 43°

> h) = 86°
> i) = 43°

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6)

- Central angle = 116°
- Angle `j` is at the circumference → half → 58°

> j) = 116° ÷ 2 = 58°

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7)

- We have angles 67°, 81°, and need k, l, m.

This is a cyclic quadrilateral with intersecting diagonals.

- Angle `k` and 67° are in the same segment → k = 67°
- Angle `l` and 81° are in the same segment → l = 81°
- Angle `m` is the remaining angle in the triangle with 67° and 81° → 180 - 67 - 81 = 32°

> k) = 67°
> l) = 81°
> m) = 32°

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8)

- Central angle = 142°
- Angle `n` is at circumference → half → 71°

> n) = 142° ÷ 2 = 71°

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9)

- Reflex central angle = 232° → so minor arc = 360° - 232° = 128°
- Angle `p` is at the circumference, subtended by the minor arc → half of 128° = 64°

> p) = 64°

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10)

- Central angle = 76°
- Angle `q` is at circumference → half → 38°

> q) = 38°

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11)

- Central angle = 122°
- Angle `r` is at circumference → half → 61°

> r) = 61°

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12)

This one has intersecting chords and multiple angles.

Given:
- 44° and 25° at circumference.

We need `s` and `t`.

#### t)
- `t` is the angle between the chords → equals half the sum of the intercepted arcs.
- Arc for 44° = 88°, arc for 25° = 50°
- So `t` = (88° + 50°)/2 = 138°/2 = 69°

#### s)
- `s` is in the same segment as 25° → so s = 25°

> s) = 25°
> t) = 69°

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## 📋 Final Answers:

| Angle | Value |
|-------|--------|
| a) | 59° |
| b) | 62° |
| c) | 47° |
| d) | 24° |
| e) | 12° |
| f) | 36° |
| g) | 98° |
| h) | 86° |
| i) | 43° |
| j) | 58° |
| k) | 67° |
| l) | 81° |
| m) | 32° |
| n) | 71° |
| p) | 64° |
| q) | 38° |
| r) | 61° |
| s) | 25° |
| t) | 69° |

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✔ All problems solved using circle theorems. Let me know if you’d like diagrams or step-by-step visuals for any part!