Printable · GCSE Higher · ages 14-16

Geometric reasoning and simple proofs worksheet — GCSE Higher

Fifteen questions on "geometric reasoning and simple proofs" — DfE statement G6. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.

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Geometric reasoning and simple proofs worksheet — GCSE Higher

Maths · GCSE Higher · G6 · Geometric reasoning and simple proofs · Non-calculator · 15 questions

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1.A hexagon has interior angles of 130°, 142°, 125°, 150°, 135° and x°. Work out the value of x.
(a)142°
(b)40°
(c)38°
(d)398°
2.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Give a reason why angle F = 58°.
(a)Angles in a triangle sum to 180°
(b)Angles on a straight line sum to 180°
(c)Vertically opposite angles are equal
(d)Base angles of an isosceles triangle are equal
3.In triangle ABC, point D lies on side AB and point E lies on side AC, so that DE is parallel to BC. AD = 3 cm, DB = 6 cm and DE = 4 cm. Work out the length of BC.
(a)4 cm
(b)8 cm
(c)10 cm
(d)12 cm
4.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Work out angle F.
(a)64°
(b)32°
(c)58°
(d)122°
5.Triangle ABC is isosceles, with AB = AC. Angle B = 70°. Which reason correctly explains why angle C = 70°?
(a)vertically opposite angles are equal
(b)corresponding angles are equal
(c)base angles of an isosceles triangle are equal
(d)angles in a triangle add up to 180°
6.A ramp is built from two straight metal supports that rest on the flat ground and meet each other at the top, forming a triangle with the ground. One support meets the ground at 34°. The angle between the two supports where they meet is 71°. Work out the angle between the second support and the ground.
(a)75°
(b)105°
(c)37°
(d)146°
7.Triangle ABC is right-angled at B. Triangle DEF is right-angled at E. AB = DE = 6 cm and AC = DF = 10 cm (AC and DF are the hypotenuses of their triangles). A student says this is not enough information to prove the triangles are congruent, because only two sides are given. Which reason shows the student is wrong?
(a)AAA applies, because both triangles have a right angle
(b)The triangles cannot be proved congruent from this information
(c)RHS applies: the right angle, hypotenuse and one matching side are all equal
(d)SSS applies, because all three sides must be equal in any right-angled triangle
8.Triangles LMN and XYZ have LM = XY, MN = YZ and LN = XZ. Give a reason why triangle LMN is congruent to triangle XYZ.
(a)SSS – all three corresponding sides are equal
(b)RHS – a right angle, hypotenuse and one other side are equal
(c)SAS – two sides and the included angle are equal
(d)ASA – two angles and the included side are equal
9.Lines AB and CD are parallel. A straight line EF crosses AB at point P and crosses CD at point Q. At P, angle APE = 65°. Angle APE and angle BPQ are vertically opposite. Angle BPQ and angle PQD are co-interior (allied) angles. Work out angle PQD.
(a)65°
(b)115°
(c)25°
(d)295°
10.Triangle ABC is extended so that side BC is extended beyond C to a point D, forming exterior angle ACD. Angle BAC = 48° and angle ABC = 67°. Which single angle fact gives angle ACD = 115° in one step, without first working out the interior angle at C?
(a)angles in a triangle add up to 180°
(b)angles on a straight line add up to 180°
(c)exterior angle = sum of the two interior opposite angles
(d)exterior angle is always double the smallest angle
11.Triangle PQR has angle P = 90°. The side PQ = 9 cm and the hypotenuse QR = 15 cm. Work out the length of PR.
(a)17.5 cm
(b)24 cm
(c)12 cm
(d)6 cm
12.In parallelogram PQRS, angle P = 65°. Which reason correctly explains why angle R = 65°?
(a)opposite angles of a parallelogram are equal
(b)co-interior angles add up to 180°
(c)alternate angles are equal
(d)angles on a straight line add up to 180°
13.Two triangular offcuts of wood, PQR and STU, are cut for a construction project. PQ = 8 cm, QR = 6 cm and angle PQR = 90°. ST = 8 cm, TU = 6 cm and angle STU = 90°. A carpenter wants to check the two pieces are identical in shape and size before using them as a matching pair. Using only the measurements given, and without working out any further lengths, which condition proves that triangle PQR is congruent to triangle STU?
(a)ASA (two angles and the included side)
(b)SAS (two sides and the included angle)
(c)RHS (right angle, hypotenuse, one side)
(d)SSS (three sides)
14.A builder props a straight plank against a vertical wall to reach a window ledge. The foot of the plank is 2.1 m from the base of the wall, and the plank is 3.5 m long. Work out how high up the wall the plank reaches.
(a)4.2 m
(b)2.8 m
(c)1.4 m
(d)7.84 m
15.A vertical flagpole casts a shadow 6 m long at the same time as a 1.2 m vertical post casts a shadow 1.8 m long. The flagpole and the post are both vertical, and the triangles formed by each object, its shadow, and the sun's ray are similar. Work out the height of the flagpole.
(a)5.4 m
(b)4 m
(c)9 m
(d)7.2 m

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