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

Let's solve each problem in Section A of the "Circle Theorems (C)" worksheet using key circle geometry theorems. I'll go step by step, explaining the reasoning behind each answer.

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🔷 Key Circle Theorems Used:

1. Angle between tangent and radius:
→ The angle between a tangent and a radius at the point of contact is 90°.

2. Alternate Segment Theorem:
→ The angle between a tangent and a chord is equal to the angle in the alternate segment.

3. Angles in the same segment:
→ Angles subtended by the same arc in the same segment are equal.

4. Angle at the center = 2 × angle at the circumference:
→ The angle at the center is twice the angle at the circumference subtended by the same arc.

5. Opposite angles in a cyclic quadrilateral sum to 180°.

6. Tangents from a common external point are equal and form equal angles.

7. Angle in a semicircle is 90°.

8. Sum of angles in a triangle = 180°.

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Now let’s solve each question:

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1) Find angle a

- Tangent touches circle at one point.
- Radius is perpendicular to tangent → so angle between radius and tangent = 90°.
- Triangle has angles: 29°, 90°, and angle a.
- Sum of angles in triangle = 180°
$$
a = 180° - 90° - 29° = 61°
$$

✔ a) 61°

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2) Find angle b

- Two tangents from an external point meet the circle.
- The angle between the two tangents is 52°.
- The triangle formed by the two tangents and the line joining the center to the external point is isosceles.
- The radii to the points of tangency are perpendicular to the tangents → each forms 90° with tangent.
- So, the quadrilateral formed has angles:
- 90°, 90°, 52°, and angle at center = ?
- But we can think of the triangle made by the two tangents and the center.
- Let’s consider the triangle from the external point to the center.
- At the external point: 52°
- At each tangent point: 90°
- So, the central angle = 360° - 90° - 90° - 52° = 128°
- But we want angle b, which is the angle between the chord and the tangent.
- This is where Alternate Segment Theorem applies.

Wait — actually, angle b is not the angle between tangent and chord? Let's look again.

Actually, angle b is inside the circle, formed by a chord and a tangent.

So, angle b is the angle between the tangent and the chord.

By Alternate Segment Theorem:
→ Angle between tangent and chord = angle in the alternate segment.

But we don’t see another angle marked yet.

Wait — the triangle outside has 52°, and it’s touching the circle via tangents.

Better approach:

Let’s consider the triangle formed by the external point and the two points of tangency.

Let’s call the external point P, and the two points of tangency A and B.

Then:
- PA and PB are tangents → PA = PB
- ∠APB = 52°
- ∠PAB = ∠PBA = ? → let’s find them.

In triangle PAB:
$$
\angle PAB + \angle PBA + 52° = 180°
\Rightarrow 2x + 52° = 180° \Rightarrow x = 64°
$$

So ∠PAB = 64°

But ∠PAB is the angle between tangent and chord AB.

By Alternate Segment Theorem, this equals the angle in the alternate segment.

So angle b (which is in the alternate segment) = 64°

✔ b) 64°

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3) Find angles c and d

We have a circle with a triangle inscribed.

Given:
- One angle at circumference = 62°
- Another = 85°
- We need to find c and d

This looks like a cyclic quadrilateral?

Wait — only three points shown. But there are two angles on the circumference.

Let’s analyze:

There is a triangle with two known angles: 62° and 85°.

Wait — no: the figure shows two chords meeting at a point on the circle.

Actually, we have:

- An inscribed triangle with angles at the circumference.
- Two angles are given: 62° and 85°
- These are both angles at the circumference.

But wait — they appear to be adjacent angles on a straight line?

No — look carefully:

The bottom side is a straight line, with angles 62° and 85° touching it.

Wait — the base is a straight line, and the two angles are at the ends of a chord.

Actually, the angle between a chord and a tangent is given?

Wait — the bottom line appears to be tangent to the circle.

Yes! There’s a tangent at the bottom.

So, angle between tangent and chord = 62° and 85°?

But that would mean two different chords from the same point?

Wait — likely, the tangent touches at one point, and two chords go up from that point.

So, the angle between tangent and chord is 62°, and another angle between tangent and other chord is 85°.

Then, by Alternate Segment Theorem:

- The angle in the alternate segment for 62° chord is 62°
- For 85° chord, alternate segment angle is 85°

So:
- Angle c is opposite to 85° — but is it in the alternate segment?

Let’s label:

- Suppose the tangent touches at point T.
- Chord TA makes 62° with tangent → then angle in alternate segment = 62°
- Chord TB makes 85° with tangent → angle in alternate segment = 85°

Now, angle c is at the top of the triangle — likely the angle subtended by the arc not including those.

Wait — actually, angle c and d are both angles in the triangle.

Let’s suppose the triangle has vertices A, B, C on the circle.

At point A: angle between tangent and chord AB is 62° → so angle in alternate segment (at B) is 62°

Similarly, at point B: angle between tangent and chord BA is 85° → so angle at A is 85°?

But that can't be — unless both angles are at different points.

Wait — maybe the tangent touches at point A, and chords go to B and C.

Then:
- Angle between tangent and chord AB = 62° → so angle ACB = 62° (alternate segment)
- Angle between tangent and chord AC = 85° → so angle ABC = 85°

Then, in triangle ABC:
- ∠ACB = 62°
- ∠ABC = 85°
- So ∠BAC = 180° - 62° - 85° = 33°

But we need to identify which is c and d.

Looking at diagram:
- Angle c is at the top — likely ∠BAC = 33°
- Angle d is at the left — likely ∠ABC = 85°

Wait — but angle d is labeled near the 62°, so perhaps:

Wait — better: the angle d is adjacent to 62° — but if 62° is between tangent and chord, and d is in the alternate segment, then d = 62°

Similarly, c is in the alternate segment for 85° → c = 85°

But that doesn't make sense because they’re both in the same triangle.

Wait — actually, both angles c and d are angles in the triangle.

From Alternate Segment Theorem:
- The angle between tangent and chord AB is 62° → so angle in alternate segment (opposite side) = 62° → that’s angle d
- Similarly, angle between tangent and chord AC is 85° → so angle in alternate segment = 85° → that’s angle c

So:
- d = 62°
- c = 85°

But then sum of angles in triangle = 62° + 85° + third angle = 147° → third angle = 33°, which is fine.

But wait — the two angles at the base are both on the tangent — so the tangent touches at the base vertex, and two chords go to the top.

So yes:
- Angle between tangent and chord to left = 62° → alternate segment angle = 62° → that’s d
- Angle between tangent and chord to right = 85° → alternate segment angle = 85° → that’s c

✔ c) 85°
✔ d) 62°

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4) Find angle e

- Tangent touches circle at one point.
- Central angle = 138°
- We need angle e at the external vertex.

This is a triangle formed by two tangents from an external point.

The two tangents touch the circle at two points, and the central angle between those points is 138°.

Let’s recall:
- The angle between two tangents from an external point is related to the central angle.

Formula:
$$
\text{Angle between tangents} = 180° - \text{central angle}
$$
Wait — no.

Actually, the quadrilateral formed by the two radii and two tangents has:
- Two 90° angles (radius ⊥ tangent)
- Central angle = 138°
- So angle at external point = 360° - 90° - 90° - 138° = 42°

So angle e = 42°

✔ e) 42°

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5) Find angle f

We have a circle with a triangle inside.

Given:
- One angle = 107° (at circumference)
- Another angle = 34° (between tangent and chord)

We need f — angle in the alternate segment?

Wait — 34° is between tangent and chord → so by Alternate Segment Theorem, the angle in the alternate segment is also 34°.

So f = 34°

✔ f) 34°

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6) Find angle g

We have a triangle with a circle inscribed? Or a tangent?

Wait — the circle has a tangent, and angle at the circumference is 123°.

But 123° is greater than 90° — so likely it’s the angle between two tangents or something.

Wait — the diagram shows a triangle with a circle inside, and a tangent.

But more clearly: the angle at the circle is 123° — but it’s at the point where a tangent meets a chord.

Wait — 123° is not the angle between tangent and chord — it's the angle at the center?

Wait — no, the dot is not at center.

Wait — the angle 123° is at the circumference, between two chords?

Wait — actually, the diagram shows:

- A triangle with a circle touching two sides.
- One angle at the vertex is g
- The angle at the circle is 123° — but it’s marked as an angle between two chords?

Wait — looking closely: the angle 123° is at the point of tangency — but it's formed by two lines: one tangent, one chord.

So, angle between tangent and chord = 123°

But that’s impossible — because the tangent is perpendicular to radius, and angles in triangle are less than 180°.

Wait — 123° is large — but possible.

But the Alternate Segment Theorem says: angle between tangent and chord = angle in alternate segment.

So, angle g is in the alternate segment → so g = 123°

But wait — angle g is at the vertex of the triangle — is it in the alternate segment?

Yes — if the tangent is on one side, and chord goes to the vertex, then angle g should be equal to the angle between tangent and chord.

But 123° seems very large — but possible.

But wait — the angle at the point of tangency is 123° — but that's not the angle between tangent and chord — it's the angle inside the circle?

Wait — actually, the angle 123° is not the angle between tangent and chord — it's the angle at the circumference, formed by two chords.

Wait — no — the diagram shows a tangent and a chord meeting at a point, forming 123°.

But that’s the angle between the tangent and chord — so by Alternate Segment Theorem, the angle in the alternate segment is 123°.

So g = 123°

But that would make angle g = 123° — and if it’s part of a triangle, the other angles must be small.

But the diagram shows angle g at the vertex, and the circle is tangent to two sides.

So yes — angle g is in the alternate segment → so g = 123°

Wait — but 123° is already the angle between tangent and chord — so alternate segment angle = 123° → so g = 123°

But that would mean the triangle has angle 123° — possible.

But wait — the angle between tangent and chord is 123°, and g is in the alternate segment → so g = 123°

✔ g) 123°

Wait — but that seems odd — usually these are acute.

Wait — could it be that the angle is reflex? No — 123° is obtuse.

But the theorem holds even for obtuse angles.

So yes — g = 123°

Wait — but let’s double-check.

Alternate Segment Theorem:
> The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

So yes — if angle between tangent and chord is 123°, then angle in alternate segment = 123° → so g = 123°

✔ g) 123°

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7) Find angle h

We have a circle with a tangent.

Given: central angle = 106°

We need angle h — between tangent and chord.

But the chord is from the point of tangency to the center?

Wait — no — the chord is from the point of tangency to another point on the circle.

Wait — the diagram shows a central angle of 106°, and a tangent at one end of the chord.

So, let’s say:
- O is center
- A is point of tangency
- B is another point on circle
- OA is radius, AB is chord
- Tangent at A
- ∠AOB = 106°

We need angle h = angle between tangent and chord AB.

We know:
- Radius OA ⊥ tangent → so angle between OA and tangent = 90°
- In triangle OAB: OA = OB (radii), so isosceles
- ∠AOB = 106° → so base angles = (180° - 106°)/2 = 37°
- So ∠OAB = 37°
- Now, angle between tangent and chord AB = angle between tangent and AB
- Since OA ⊥ tangent, and ∠OAB = 37°, then angle between tangent and AB = 90° - 37° = 53°

So h = 53°

✔ h) 53°

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8) Find angle i

We have a circle with a triangle inscribed.

One angle outside = 72° — that’s an external angle.

Also, the triangle has two equal sides marked with ticks — so isosceles triangle.

We need angle i at the top.

But first, note: the triangle is inscribed in the circle, and the side of the triangle is tangent to the circle? No — the triangle is circumscribed around the circle?

Wait — no — the circle is inscribed in the triangle? But the triangle is drawn around the circle, and the circle touches all three sides.

But the angle i is at the top of the triangle, and we have a 72° angle at the bottom left.

But the circle is tangent to the sides.

So this is a triangle with an incircle.

But angle i is the angle at the top.

But how do we find it?

Wait — the angle i is at the vertex, and we have a 72° angle at the bottom left.

But we need more.

Wait — the circle is tangent to the sides, and the triangle has two equal sides (ticks), so it’s isosceles.

So base angles are equal.

But we are given one base angle = 72° → so the other base angle = 72°

Then apex angle i = 180° - 72° - 72° = 36°

✔ i) 36°

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9) Find angles j and k

We have a triangle with a circle, tangents.

Given:
- One angle in the circle = 34°
- Another = 41°
- Need j and k

First, angle j is inside the circle — likely in a triangle.

Note: 34° and 41° are angles between tangents and chords.

So by Alternate Segment Theorem:

- Angle between tangent and chord = angle in alternate segment

So:
- The 34° angle is between tangent and chord → so angle in alternate segment = 34° → that’s angle j
- The 41° angle is between tangent and chord → so angle in alternate segment = 41° → that’s angle k

Wait — but angle j is marked at the top — is it in the alternate segment?

Let’s assume:
- The tangent at one point makes 34° with chord → so angle in alternate segment = 34° → that’s j
- The tangent at another point makes 41° with chord → so angle in alternate segment = 41° → that’s k

But k is at the bottom — so yes.

So:
- j = 34°
- k = 41°

But wait — is that correct?

Alternatively, perhaps j is the angle at the circumference subtended by the same arc.

But from the diagram:
- 34° is between tangent and chord → so alternate segment angle = 34° → that’s j
- 41° is between tangent and chord → alternate segment angle = 41° → that’s k

Yes.

✔ j) 34°
✔ k) 41°

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✔ Final Answers:

| Question | Answer |
|--------|--------|
| 1) a | 61° |
| 2) b | 64° |
| 3) c | 85° |
| 3) d | 62° |
| 4) e | 42° |
| 5) f | 34° |
| 6) g | 123° |
| 7) h | 53° |
| 8) i | 36° |
| 9) j | 34° |
| 9) k | 41° |

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📌 Summary of Theorems Used:

- Tangent-Radius Perpendicular: 90° at point of contact.
- Alternate Segment Theorem: Angle between tangent and chord = angle in alternate segment.
- Isosceles Triangles: Equal sides → equal angles.
- Sum of angles in triangle: 180°.
- Quadrilateral angles: 360°.
- Central angle = 2 × circumference angle (used indirectly).

Let me know if you'd like diagrams or further explanation!