Geometric reasoning and simple proofs worksheet — GCSE Foundation

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

2.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.

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

4.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.

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

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

8.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.

9.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?

10.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.

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

12.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.

13.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.

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

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