GCSE Maths Angles & constructions questions by topic

Point-level angle facts.

1. (a)Two angles on a straight line: one is 118°. Find the other.[1]
2. (b)Angles of 210° and 55° meet at a point with a third angle y. Find y.[2]
3. (c)Two straight lines cross making an angle of 46°. Write down the angle vertically opposite it.[1]

Parallel-line rules.

1. (a)A transversal crosses parallel lines making 71° with the first. Write down the corresponding angle on the second.[1]
2. (b)Write down the alternate angle to 128° between the same parallels.[1]
3. (c)Find the co-interior partner of 97°, and state the rule you used.[2]

Triangle angles.

1. (a)A triangle has angles 54° and 67°. Find the third angle.[2]
2. (b)An isosceles triangle has base angles of 32°. Find its apex angle.[2]

Quadrilaterals.

1. (a)Three angles of a quadrilateral are 100°, 75° and 88°. Find the fourth.[2]
2. (b)In parallelogram ABCD, angle A = 115°. Find angles C and B.[2]

A clock face is divided evenly.

1. (a)A clock face has 12 equal sectors. Work out the angle of one sector.[2]
2. (b)Work out the angle between the hour hand and the minute hand at 2 o’clock.[2]

A transversal and a triangle share a diagram.

1. (a)Parallel lines PQ and RS are cut by a transversal. One angle is 72°. Work out its co-interior partner.[2]
2. (b)A triangle in the same diagram has two angles of 72° and 63°. Work out the third angle.[2]

Angle algebra.

1. (a)Two angles on a straight line are (2x)° and (3x)°. Find x and both angles.[2]
2. (b)Four angles meet at a point: x°, 2x°, 3x° and 120°. Find x and the three unknown angles.[3]

Regular polygons.

1. (a)Find the interior angle of a regular octagon.[2]
2. (b)A regular polygon has interior angles of 162°. How many sides does it have?[3]

Exterior angles.

1. (a)A triangle has interior angles 48° and 61°. Find the exterior angle at the third vertex.[2]
2. (b)A triangle’s exterior angle is (5x)° and the opposite interior angles are (x + 20)° and (2x + 30)°. Find x and all three interior angles.[3]

Constructing a perpendicular bisector.

1. (a)Describe the steps to construct the perpendicular bisector of a segment AB using ruler and compasses.[2]
2. (b)A point P is the same distance from A as from B. State which line P must lie on.[1]
3. (c)Explain why the compass radius must stay the same when drawing the arcs from A and from B.[2]

A regular pentagon is examined.

1. (a)Work out the interior angle sum of a pentagon.[2]
2. (b)Use your answer to part (a) to work out the interior angle of a regular pentagon.[1]
3. (c)Work out one exterior angle, giving a reason.[2]

Proving the exterior angle theorem. (Higher)

1. (a)Prove that the exterior angle of a triangle equals the sum of the two opposite interior angles. (Draw a line through the third vertex parallel to the opposite side.)[3]
2. (b)A triangle’s exterior angle is 110° and the opposite interiors are (2x + 15)° and (x + 5)°. Find x and all three interior angles.[3]

A shaded loci region. (Higher)

1. (a)A rectangular field PQRS has PQ = 12 m and QR = 7 m. Describe the constructions needed to shade the region inside the field that is less than 5 m from P and closer to PQ than to SR.[3]
2. (b)Explain why the halfway line between PQ and SR may be constructed as the perpendicular bisector of a 7 m segment across the field.[2]
3. (c)Describe the final boundary of the shaded region.[1]

A regular polygon with algebra. (Higher)

1. (a)A regular polygon has interior angle (5x)° and exterior angle x°. Find x.[2]
2. (b)How many sides does the polygon have?[2]
3. (c)Explain why no regular polygon can have an exterior angle of exactly 25°.[2]

An isosceles trapezium. (Higher)

1. (a)Trapezium ABCD has AB parallel to DC and AD = BC. Angle ADC = 68°. Find angle BCD and angle DAB.[4]
2. (b)Find angle ABC and explain your reasoning.[2]

A goat is tied to a shed. (Higher)

1. (a)Describe the locus of points exactly 3 m from the straight wall AB of a shed, where AB is 6 m long.[2]
2. (b)A goat is tied by a 5 m rope to a corner of a 4 m × 3 m shed. Describe the full locus it can reach.[3]
3. (c)State how far round the shed the rope wraps on each of the two far walls.[1]

Algebra lives on parallel lines. (Higher)

1. (a)Two co-interior angles between parallel lines are (3x + 15)° and (2x + 25)°. Form and solve an equation for x.[3]
2. (b)Work out the size of each angle.[2]
3. (c)State the angle rule your equation used.[1]

The triangle angle sum is proved. (Higher)

1. (a)State the extra line you draw to prove that the angles of a triangle sum to 180°.[2]
2. (b)Complete the proof.[3]
3. (c)Hence explain why a triangle cannot contain two right angles.[1]