Geometric reasoning and simple proofs worksheet — GCSE Foundation

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

2.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.

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

4.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.

5.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.

6.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.

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

8.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.

9.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.

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

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

12.Triangle ABC is right-angled at B. Triangle DEF is right-angled at E. AB = DE = 6 cm and AC = DF = 10 cm (AC and DF are the hypotenuses of their triangles). A student says this is not enough information to prove the triangles are congruent, because only two sides are given. Which reason shows the student is wrong?

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

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

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