Circle Angles, Arcs & Chords — Challenge
Where the vertex of an angle sits — centre, edge or outside — decides which rule applies.
What these problems 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
An inscribed angle in a circle stands on a diameter. How many degrees is it?
- 45
- 60
- 90
- 120
- 180
Worked answer
90
A diameter cuts off a semicircular arc, which corresponds to a central angle of 180 degrees.
An inscribed angle is half the central angle on the same arc.
So the inscribed angle is 1802 = 90 degrees, which is the angle in a semicircle.
Problem 2
A central angle of 80 degrees stands on an arc. How many degrees is the inscribed angle on the same arc?
- 20
- 40
- 80
- 100
- 160
Worked answer
40
An inscribed angle is always half the central angle standing on the same arc.
802 = 40 degrees.
Doubling instead would give 160, which reverses the relationship.
Problem 3
A central angle measures 90 degrees. How many degrees is the arc it cuts off?
- 45
- 90
- 180
- 270
- 360
Worked answer
90
An arc is measured by the central angle that stands on it.
So a central angle of 90 degrees cuts off an arc of 90 degrees, a quarter of the circle.
Problem 4
What do opposite angles of a cyclic quadrilateral add up to, in degrees?
- 90
- 180
- 270
- 360
- 120
Worked answer
180
A cyclic quadrilateral has all four vertices on one circle.
Each pair of opposite angles stands on the two arcs that together make the whole circle, so the two inscribed angles are half of 360 degrees between them.
That gives 180 degrees for each opposite pair.
Problem 5
At how many degrees does a tangent meet the radius drawn to the point of contact?
- 30
- 45
- 60
- 90
- 180
Worked answer
90
A tangent touches the circle at exactly one point.
The radius to that point is the shortest distance from the centre to the tangent, and the shortest distance is always perpendicular.
So they meet at 90 degrees.
Problem 6
An inscribed angle measures 35 degrees. How many degrees is the arc it stands on?
- 17.5
- 35
- 70
- 105
- 140
Worked answer
70
The arc is measured by its central angle, and the inscribed angle is half of that.
So the central angle, and the arc, is 2· 35 = 70 degrees.
Halving instead would give 17.5, which runs the relationship backwards.
Problem 7
Two tangents are drawn to a circle from the same external point. How do their lengths compare?
- They are equal
- They are perpendicular
- They are parallel
- They are unequal
- They are diameters
Worked answer
They are equal
Join the external point to the centre, and join the centre to each point of contact.
This makes two right triangles sharing the same hypotenuse, with radii of equal length as one leg each.
By Pythagoras the remaining legs are equal, so the two tangent lengths are equal.
Problem 8
One angle of a cyclic quadrilateral is 110 degrees. How many degrees is the angle opposite it?
- 55
- 70
- 110
- 180
- 250
Worked answer
70
Opposite angles of a cyclic quadrilateral add to 180 degrees.
180-110 = 70 degrees.
The other pair of opposite angles must also add to 180, so all four angles total 360, as for any quadrilateral.