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

Here is the complete solution for Section A of the “Circle Theorems (C)” worksheet, with clear explanations for each problem.

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1) Angle a

Diagram: A circle with center marked. A tangent touches the circle at one point, and a radius goes to that point. An angle of 29° is formed between the tangent and a chord from the point of contact.

Theorem used: *The angle between a tangent and a chord is equal to the angle in the alternate segment.*

But wait — here we have a radius drawn to the point of contact. Important fact: A radius is perpendicular to the tangent at the point of contact.

So, the radius and the tangent form a 90° angle.

We are given an angle of 29° between the tangent and the chord. Therefore, the angle between the radius and the chord is:

> a = 90° – 29° = 61°

✔ Answer: a = 61°

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2) Angle b

Diagram: Two tangents drawn from an external point to a circle, forming a triangle with the chord connecting the two points of contact. The angle between the two tangents is 52°.

Theorem used: *Two tangents from an external point are equal in length, so the triangle formed is isosceles. Also, the angle between the two tangents and the angle subtended by the chord at the center are related.*

Alternatively, use the fact that the angle between the two tangents is 52°, and the angles at the base of the isosceles triangle (angles between tangent and chord) are equal.

In the triangle formed by the two tangents and the chord, the sum of angles is 180°.

Let each base angle be b.

> 52° + b + b = 180°
> 2b = 128°
> b = 64°

✔ Answer: b = 64°

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3) Angles c and d

Diagram: A circle with a tangent at the bottom. Two chords go from the point of contact to two points on the circumference, forming angles of 62° and 85° with the tangent.

Theorem used: *Alternate Segment Theorem* — the angle between the tangent and a chord equals the angle subtended by the chord in the alternate segment.

- The angle between the tangent and the left chord is 62° → this equals angle d (in the alternate segment).
- The angle between the tangent and the right chord is 85° → this equals angle c (in the alternate segment).

✔ Answer: c = 85°, d = 62°

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4) Angle e

Diagram: Two tangents from an external point to a circle. The angle at the center between the two radii is 138°.

Theorem used: *Angle between two tangents from an external point = 180° minus the angle at the center.*

Why? Because the quadrilateral formed by the two radii and two tangents has two right angles (radius ⊥ tangent), so:

> Sum of angles in quadrilateral = 360°
> 90° + 90° + 138° + e = 360°
> e = 360° – 318° = 42°

✔ Answer: e = 42°

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5) Angle f

Diagram: A tangent at a point on the circle, and a chord going to another point. The angle between the tangent and the chord is 34°. Another angle subtended by the same arc at the circumference is 107°. We are to find angle f, which is also at the circumference, subtended by the same arc.

Wait — let’s interpret carefully.

Actually, angle f is the angle in the alternate segment to the 34° angle.

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

So, f = 34°

But there’s also a 107° angle shown — that’s probably the reflex angle or part of a cyclic quadrilateral? Actually, looking again — the 107° is likely the angle subtended by the major arc, while f is subtended by the minor arc.

Actually, f is opposite the 107° angle in a cyclic quadrilateral? Not quite.

Re-examining: The diagram shows a tangent, a chord, and then a triangle inscribed. The 34° is between tangent and chord. The 107° is the angle at the circumference on the other side of the chord.

This suggests that f and 107° are angles subtended by the same arc? No — they’re on opposite sides.

Actually, f is the angle in the alternate segment to the 34° angle → so f = 34°

The 107° might be a red herring or misdirection — unless it’s part of a different theorem.

Wait — perhaps the 107° is the angle at the circumference subtended by the arc, and f is the angle between the chord and the tangent — but no, 34° is given as that.

Actually, I think the 107° is the angle of the triangle at the circumference, and f is the other angle in the triangle.

The triangle has angles: 34° (at tangent-chord), 107° (at circumference), and f.

Sum of angles in triangle = 180°

> f = 180° – 34° – 107° = 39°

That makes sense.

✔ Answer: f = 39°

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6) Angle g

Diagram: Two tangents from an external point to a circle. The angle at the center (between radii) is 123°.

Theorem used: Same as Q4 — angle between two tangents = 180° – angle at center.

> g = 180° – 123° = 57°

✔ Answer: g = 57°

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7) Angle h

Diagram: A tangent and a chord from the point of contact. The angle at the center subtended by the chord is 106°.

Theorem used: *Angle between tangent and chord = half the difference of the intercepted arcs?*

Wait — better: The angle between the tangent and the chord is equal to the angle in the alternate segment.

But we don’t have that directly.

Alternatively: The angle between the tangent and the chord = half the angle at the center (if it’s the angle subtended by the same arc).

Actually, no — the angle between tangent and chord is equal to the angle subtended by the chord in the alternate segment.

But we can relate it to the central angle.

Recall: The angle subtended at the circumference is half the angle at the center.

So, the angle in the alternate segment (which equals angle h) is half of 106°.

> h = 106° ÷ 2 = 53°

✔ Answer: h = 53°

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8) Angle i

Diagram: A circle with a triangle inscribed. Two sides are marked equal (isosceles triangle). The angle at the top vertex outside the circle is 72° — actually, it's the angle between the two tangents.

Wait — the 72° is the angle between the two tangents from an external point.

The triangle inside is isosceles (two sides equal), and inscribed.

Actually, the triangle is formed by two tangents and a chord — and the two tangent segments are equal, so the triangle is isosceles.

The angle at the external point is 72°, so the base angles are equal.

> i = (180° – 72°) ÷ 2 = 108° ÷ 2 = 54°

✔ Answer: i = 54°

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9) Angles j and k

Diagram: Two tangents from an external point to a circle. The angle between one tangent and a chord is 34°, and between the other tangent and the other chord is 41°. We need to find angle j (angle in alternate segment to 34°) and angle k (the angle between the two tangents).

First, angle j:

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

So, j = 34°

Now, angle k:

The angle between the two tangents = 180° – (angle subtended by the arc between the two points of contact at the center).

But we can also find it using the triangle formed.

The triangle formed by the two tangents and the chord has angles:

- At the external point: k
- At the two base vertices: angles between tangent and chord → 34° and 41°

Wait — no, those 34° and 41° are angles between the tangents and the chords — which are equal to the angles in the alternate segments (like j).

But in the triangle formed by the two tangents and the chord, the angles at the circle are j and another angle (let’s say m).

Actually, the two angles at the base of the triangle (where the tangents meet the circle) are equal to the angles in the alternate segments — so 34° and 41°.

Therefore, in the triangle:

> k + 34° + 41° = 180°
> k = 180° – 75° = 105°

✔ Answer: j = 34°, k = 105°

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## ✔ Final Answers:

1) a = 61°
2) b = 64°
3) c = 85°, d = 62°
4) e = 42°
5) f = 39°
6) g = 57°
7) h = 53°
8) i = 54°
9) j = 34°, k = 105°

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