Angle facts

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

AB, BC, CD and DE are four sides of a regular polygon with n sides.

BCX is an equilateral triangle. CDYX is a square. Work out the value of n. You must show all your working.

AB, BC, CD and DE are four sides of a regular polygon with n sides.

BCX is an equilateral triangle. CDYX is a square. Work out the value of n. You must show all your working.

PQR is a straight line.

Work out the value of x.

Give a reason for your answer.

ABCD is a quadrilateral. BCF and DCE are straight lines.

Work out the size of the angle marked x. Give a reason for each stage of your working.

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

Find the size of angle OBC. Write down any circle theorems that you use.

ABCD is a parallelogram.

All angles are measured in degrees. Find an expression, in terms of e, for the size of angle CAD. Give a reason for each stage of your working.

Here is a pentagon.

Angle a = angle c Angle b = 155° Angle d is three times the size of angle c Angle e is two times the size of angle c Work out the size of angle a

ABCD is a parallelogram.

All angles are measured in degrees. Find an expression, in terms of e, for the size of angle CAD. Give a reason for each stage of your working.

Here is a pentagon.

Angle a = angle c Angle b = 155° Angle d is three times the size of angle c Angle e is two times the size of angle c Work out the size of angle a

ABC is a triangle.

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

The diagram shows two congruent regular 9-sided polygons. ABCD is a quadrilateral.

Show that ABCD is not a square.

OA, OB and OC are three straight lines.

Work out the size of the angle marked x.

Give a reason for your answer.

ABCD is a quadrilateral.

All angles are measured in degrees. Show that ABCD is a trapezium.

ABCD is a quadrilateral.

All angles are measured in degrees. Show that ABCD is a trapezium.

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.

The diagram shows a quadrilateral ABDE and an equilateral triangle BCD.

CB is parallel to DE. Angle AED = 132° Work out the size of the angle marked x. You must give a reason for each stage of your working.

ABCD is a trapezium.

AD is parallel to BC. Calculate the area of triangle BCD. Give your answer correct to 1 decimal place.

A, B, C and D are points on the circumference of a circle, centre O. AC is a diameter of the circle. ADE and BCE are straight lines. Work out the size of angle BDC. Write down any circle theorems that you use.

Work out the size of the angle marked x.

A student says that an angle of 50° is an obtuse angle. The student is wrong.

Explain why.

Here is a pentagon.

Angle AED = 4×angle ABC Work out the size of angle AED. You must show all your working.

ABC is a triangle.

Calculate the size of angle BAC. Give your answer correct to 1 decimal place.

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.

Find the value of y.

Give a reason for your answer.

ABCD is a quadrilateral.

Work out the size of angle x.

Give a reason for your answer.

The diagram below shows a triangle.

The diagram is wrong.

Explain why.

ABDE is a cyclic quadrilateral. ABC and EDC are straight lines. Angle DBC = 60° Given that size of angle EAB: size of angle BCD = 2: 1 work out the size of angle BCD. You must show all your working.

ABCDEFG is a regular heptagon.

The area of triangle ABG is 30 cm² Calculate the length of GB. Give your answer correct to 3 significant figures. You must show all your working.

Here is a regular hexagon and a regular pentagon.

Work out the size of the angle marked x. You must show all your working.

Here is a regular hexagon and a regular pentagon.

Work out the size of the angle marked x. You must show all your working.

The diagram shows a triangle ABC.

ACD and BCE are straight lines. Work out the size of the angle marked x. Give a reason for each stage of your working.

A, B, C and D are points on the circumference of a circle, centre O. ADE and BCE are straight lines.

Work out the size of angle CDE. Give a reason for each stage of your working.

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.

RST is a straight line.

Work out the value of x.

Give a reason for your answer.

A, B, C and D are four points on a circle. SBT is a tangent to the circle. Angle ABD = 20 the size of angle BAD: the size of angle BCD = 3: 1 Find the size of angle SBA. Give a reason for each stage of your working.

Jenna measures all the angles around a point. Her results are 23, 145, 23 and 69 Explain why these results cannot be true.

ABC is a triangle.

D is the point on BC such that angle BAD = angle DAC = x°

ADC is a triangle.

EB is parallel to DC. Angle EBC = 148 Angle ADC = 63 Work out the size of angle EAB. You must give a reason for each stage of your working.

ADC is a triangle.

EB is parallel to DC. Angle EBC = 148 Angle ADC = 63 Work out the size of angle EAB. You must give a reason for each stage of your working.

ABC is a straight line.

Work out the size of the angle marked x.

Give a reason for your answer.

The diagram below is wrong.

Explain why.

28Each exterior angle of a regular polygon is 15 Work out the number of sides of the polygon.

PQRS is a quadrilateral. PST is a straight line.

Find the value of y.

Points B, D, E and F lie on a circle. ABC is the tangent to the circle at B. Find the size of angle ABD. You must give a reason for each stage of your working.

PQR is a straight line.

The pie charts show information about the favourite animal of each student at school A and of each student at school B.

There are 480 students at school A. There are 760 students at school B.

Henry says,

“The same number of students at each school have tigers as their favourite animal.”

Is Henry correct? You must show how you get your answer.

The diagram shows triangle ABC.

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

Work out the size of angle BDC.

The diagram shows a regular pentagon and a parallelogram.

Work out the size of the angle marked x. You must show all your working.

The diagram shows a regular pentagon and a parallelogram.

Work out the size of the angle marked x. You must show all your working.

The diagram shows triangle ABC.

AB = 3.4 cm AC = 6.2 cm BC = 6.1 cm

D is the point on BC such that

Calculate the length DC. Give your answer correct to 3 significant figures. You must show all your working.

AB and BC are perpendicular lines.

Find the value of x.

RS and TU are parallel lines. PQ is a straight line.

An angle of size 125 is shown on the diagram.

Write down the letter of one other angle of size 125 Give a reason for your answer.

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 a hexagon. The hexagon has one line of symmetry.

117°

Angle ABC 117° Angle BCD 2 angle CDE Work out the size of angle AFE. You must show all your working.

The diagram shows a hexagon. The hexagon has one line of symmetry.

117°

Angle ABC 117° Angle BCD 2 angle CDE Work out the size of angle AFE. You must show all your working.

The diagram shows the positions of three towns, Acton (A), Barston (B) and Chorlton (C).

Chorlton is 9 km from Barston on a bearing of 150° Find the bearing of Chorlton from Acton. Give your answer correct to 1 decimal place. You must show all 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.

The size of each interior angle of a regular polygon is 11 times the size of each exterior angle.

Work out how many sides the polygon has.

BEG is a triangle.

ABC and DEF are parallel lines.

Work out the size of angle x. Give a reason for each stage of your working.

BEG is a triangle.

ABC and DEF are parallel lines.

Work out the size of angle x. Give a reason for each stage of your working.

ABC is a right-angled triangle.

AC = 14 cm. Angle C = 90

size of angle B: size of angle A = 3: 2

Work out the length of AB. Give your answer correct to 3 significant figures.

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.

ABCDE is a pentagon.

Angle BCD = 2 angle ABC

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

ABCDE is a pentagon.

Angle BCD = 2 angle ABC

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

ABCD is a parallelogram. EDC is a straight line. F is the point on AD so that BFE is a straight line.

Angle EFD = 35 Angle DCB = 75

Show that angle ABF = 70 Give a reason for each stage of your working.

ABCD is a parallelogram. EDC is a straight line. F is the point on AD so that BFE is a straight line.

Angle EFD = 35 Angle DCB = 75

Show that angle ABF = 70 Give a reason for each stage of your working.

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.

The size of the largest angle in a triangle is 4 times the size of the smallest angle. The other angle is 27° less than the largest angle.

Work out, in degrees, the size of each angle in the triangle. You must show your working.

In the diagram, AB, BC and CD are three sides of a regular polygon P.

Show that polygon P is a hexagon. You must show your working.

In the diagram, AB, BC and CD are three sides of a regular polygon P.

Show that polygon P is a hexagon. You must show your working.