Geometric reasoning and simple proofs worksheet — GCSE Foundation

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

2.Triangle PQR has angle P = 90°. The side PQ = 9 cm and the hypotenuse QR = 15 cm. Work out the length of PR.

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

4.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.

5.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.

6.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Give a reason why angle F = 58°.

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

8.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.

9.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.

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

11.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.

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

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

14.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.

15.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?