Geometric reasoning and simple proofs worksheet — GCSE Foundation

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

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

3.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?

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

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

6.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.

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

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

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

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

11.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.

12.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.

13.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?

14.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.

15.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.