Geometric reasoning and proof

AED and BCD are straight lines. ED = EC Show that EC is parallel to AB. Give a reason for each stage of your working.

ABC is a right-angled triangle.

AEC and ADB are straight lines. ED is parallel to CB.

Prove that triangle ABC is similar to triangle ADE.

ED = 20 cm CB = 30 cm AC = 18 cm

Work out the length of EC.

ABC is a triangle.

BCD is a straight line. Show that triangle ABC is isosceles. Give a reason for each stage of your working.

A, B and C are three points on a circle, centre O.

BA = BC Prove that OB bisects angle ABC.

ABC and BCD are isosceles triangles.

AB = BC = CD Angle CAB = 81 Angle BCD = 4×angle ABC Find the size of angle ABC: the size of angle CBD Give your answer in the form 1:n You must show all your working.

The diagram shows an isosceles triangle ABD and the straight line ABC.

BA = BD x:y = 2: 1 Work out the value of w.

The diagram shows an isosceles triangle ABD and the straight line ABC.

BA = BD x:y = 2: 1 Work out the value of w.

ACF and ADG are straight lines. BCD and EFG are parallel lines.

Show that triangle ACD is isosceles. Give a reason for each stage of your working.

ACF and ADG are straight lines. BCD and EFG are parallel lines.

Show that triangle ACD is isosceles. Give a reason for each stage of your working.

The diagram shows a triangle ADE.

AE = DE AB:BC:CD = 1: 2: 1 Prove that triangle ACE is congruent to triangle DBE.

In the diagram, PQR is an isosceles triangle with PQ = PR.

APR and CQD are parallel lines. BPQ is a straight line. Angle APB = 56° Work out the size of angle CQR. Give a reason for each stage of your working.

P, Q, R and S are four points on a circle.

PXR and SXQ are straight lines. Prove that triangle PQX and triangle SRX are similar.

the size of angle DCB: the size of angle ACD = 2: 1

Work out the size of angle BDC.

The diagram shows a right-angled triangle.

All the measurements are in centimetres.

The area of the triangle is 27.5 cm²

Work out the length of the shortest side of the triangle. You must show all your working.

The points A, B, C and D lie on a circle. CDE is a straight line.

Angle ABD = 40° Work out the size of angle ADE. You must give a reason for each stage of your working.

The diagram shows quadrilateral ABCD with each of its sides extended.

Show that ABCD is a kite. Give a reason for each stage of your working.

Mary needs to work out the size of angle x in this diagram.

She writes x = 63 because base angles of an isosceles triangle are equal.

Mary is wrong.

Explain why.

William needs to work out the size of angle y in this diagram.

William writes

Working	Reason
angle EGH = 57	because corresponding angles are equal
y = 180 – 57 y = 123	because angles on a straight line add up to 180

One of William’s reasons is wrong.

Write down the correct reason.

ABCD is a parallelogram. ABP and QDC are straight lines. Angle ADP = angle CBQ = 90

Prove that triangle ADP is congruent to triangle CBQ.

Explain why AQ is parallel to PC.

12

RS and ST are 2 sides of a regular 12-sided polygon. RT is a diagonal of the polygon.

Work out the size of angle STR. You must show your working.

BCD is a straight line. ABC is a triangle.

Show that triangle ABC is an isosceles triangle. Give a reason for each stage of your working.

A, B and C are points on the circumference of a circle, centre O. AOB is a diameter of the circle.

Prove that angle ACB is 90° You must not use any circle theorems in your proof.

A, B, C and D are four points on the circumference of a circle.

AEC and BED are straight lines.

Prove that triangle ABE and triangle DCE are similar. You must give reasons for each stage of your working.