Geometric reasoning and simple proofs worksheet — GCSE Foundation

1.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.

2.Two angles lie on a straight line. One angle is 118°. Which reason correctly explains why the other angle is 62°?

3.Triangle XYZ has angle X = 90°. Angle Y = 34°. Work out angle Z.

4.Three angles meet at a single point on one side of a straight line. Two of the angles are 48° and 65°. Work out the third angle.

5.Two straight lines AB and CD cross at point O. Angle AOC = 74°. Work out angle AOD.

6.Two straight lines PQ and RS cross at point O. Angle POR = 52°. Work out angle QOS.

7.A children's slide is built as a right-angled triangle: the vertical ladder side is 2.4 m, the horizontal base along the ground is 3.2 m, and the sloping slide is the third side. Work out the exact length of the sloping slide.

8.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.

9.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.

10.A hexagon has interior angles of 130°, 142°, 125°, 150°, 135° and x°. Work out the value of x.

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

12.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.

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

14.Triangle ABC is similar to triangle PQR. AB corresponds to PQ, and BC corresponds to QR. AB = 5 cm, PQ = 15 cm and BC = 7 cm. Work out the length of QR.

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.