Geometric reasoning and simple proofs worksheet — GCSE Foundation

1.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.

2.Triangle ABC is isosceles, with AB = AC. Angle B = 70°. Which reason correctly explains why angle C = 70°?

3.A quadrilateral has angles of 100°, 85° and 95° at three of its vertices. Work out the fourth angle.

4.In parallelogram PQRS, angle P = 65°. Which reason correctly explains why angle R = 65°?

5.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.

6.Lines PQ and RS are parallel. A straight line crosses PQ at point X and crosses RS at point Y. Angle PXY = 63°. Angle PXY and angle XYS are co-interior (allied) angles, which add up to 180°. Work out the size of angle XYS.

7.A roof truss is shaped like an isosceles triangle resting on a horizontal ceiling joist. The truss's sloping edges are equal in length, and the angle at its apex is 40°. The ceiling joist continues in a straight line beyond the foot of the truss. Work out the angle between the truss's sloping edge and the extended joist, on the outside of the truss.

8.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?

9.Triangles LMN and XYZ have LM = XY, MN = YZ and LN = XZ. Give a reason why triangle LMN is congruent to triangle XYZ.

10.Lines JK and LM are parallel. A straight line crosses JK at point P and crosses LM at point Q. Angle KPQ = 118°. Angle KPQ and angle MQP are co-interior (allied) angles. Work out angle MQP.

11.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Work out angle F.

12.Two angles lie on a straight line. One angle is 37°. Work out the size of the other angle.

13.A straight line passes through points A, B and C, with B between A and C. A fourth point D is not on the line. Angle ABD = 132°. Work out the size of angle DBC.

14.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.

15.A rectangular gate is braced with a diagonal strut. The gate is 1.2 m wide and 0.9 m tall. Work out the exact length of the diagonal strut.