Geometric reasoning and simple proofs worksheet — GCSE Foundation

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

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

3.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.

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

5.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.

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

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

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.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?

10.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.

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

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

14.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?

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