Circle Angles, Arcs & Chords — Practice
Where the vertex of an angle sits — centre, edge or outside — decides which rule applies.
What these questions cover
These questions use 11 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Supplementary — Two angles summing to 90° / to 180°.
- Radius — Centre to edge. Half the diameter.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Central angle — An angle whose vertex is the centre. Equal to its arc.
- Inscribed angle — An angle whose vertex is on the circle. Half its arc.
- Chord — A segment joining two points on the circle.
- Tangent — A line touching the circle at exactly one point.
- Secant — A line that cuts the circle at two points.
Problem 1
A central angle cuts an arc of 70°. Find the angle.
Worked answer
70
70°
Problem 2
An inscribed angle cuts an arc of 86°. Find the angle.
Worked answer
43
86 ÷ 2 = 43°
Problem 3
An angle is inscribed in a semicircle. Find its measure.
Worked answer
90
The arc is 180°.
Half is 90°.
Problem 4
An inscribed angle measures 35°. Find its arc.
Worked answer
70
2 × 35 = 70°
Problem 5
An angle with its vertex at the centre equals
- its arc
- half its arc
- twice its arc
- half the difference
Worked answer
its arc
At the centre, so it equals the arc.
Problem 6
Two secants from an external point cut arcs 120° and 46°. Find the angle.
Worked answer
37
120 - 46 = 74
74 ÷ 2 = 37°
Problem 7
Two chords cross inside a circle cutting arcs 80° and 40°. Find the angle.
Worked answer
60
80 + 40 = 120
120 ÷ 2 = 60°
Problem 8
A tangent touches a circle of radius 9 at T. From P, OP = 15. Find PT.
Worked answer
12
PT2 = 152 - 92 = 144
PT = 12
Problem 9
Two chords cross so the pieces are 3, 8 and 4, x. Find x.
Worked answer
6
3 × 8 = 4x
24 = 4x
x = 6
Problem 10
Two inscribed angles stand on the same arc. One is 28°. Find the other.
Worked answer
28
Angles on the same arc are congruent.
28°
Problem 11
A tangent meets the radius at the point of contact at
- 90°
- 45°
- 180°
- any angle
Worked answer
90°
A tangent is perpendicular to the radius there.
Problem 12
A central angle is 124°. Find the inscribed angle on the same arc.
Worked answer
62
124 ÷ 2 = 62°
Problem 13
In a cyclic quadrilateral one angle is 105°. Find the opposite angle.
Worked answer
75
180 - 105 = 75°
Problem 14
Two tangents from an external point cut arcs of 210° and 150°. Find the angle at the point.
Worked answer
30
210 - 150 = 60
60 ÷ 2 = 30°
Problem 15
An inscribed angle is (3x)° and its arc is (8x - 20)°. Find x.
Worked answer
10
8x - 20 = 2(3x)
8x - 20 = 6x
x = 10
Problem 16
Two tangents are drawn from an external point. One has length 3x + 2, the other 5x - 8. Find x.
Worked answer
5
3x + 2 = 5x - 8
10 = 2x
x = 5