Congruence, Similarity & Geometrical Proof (AQA GCSE Maths)

Exam Questions

3 hours50 questions
1a
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3 marks

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ABC is an equilateral triangle.
D lies on BC.
AD is perpendicular to BC.

Prove that triangle ADC is congruent to triangle ADB.

1b
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2 marks

Hence, prove that B D space equals 1 half A B.

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2
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4 marks

A B C D is a quadrilateral.

q1-easy-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

A B space equals space C D.
Angle A B C = angle B C D.

Prove that A C equals space B D.

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3a
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2 marks

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Quadrilaterals ABCD and LMNP are mathematically similar.

Angle A = angle L
Angle B = angle M
Angle C = angle N
Angle D = angle P

Work out the length of LP.

3b
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2 marks

Work out the length of BC.

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4
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3 marks

A B C and A B D are two right-angled triangles.

q4-easy-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

Angle space B A C space equals space angle space A D B space equals space 90 degree

A B space equals space 13 space cm space
D B space equals space 5 space cm

Work out the length of C B.

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5
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1 mark

q4-paper-2h-nov-2021-aqa-gcse-maths

Circle the reason why these triangles are congruent.

ASA          RHS   SAS    SSS

     

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6
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1 mark

q2-paper-2h-nov-2020-aqa-gcse-maths

Circle the reason why these triangles are congruent.

RHS ASA SSS SAS

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7
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2 marks

These two triangles are similar.

q9-paper-3h-nov-2020-aqa-gcse-maths

Work out the value of a.

............................cm

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8
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1 mark

Here are two right-angled triangles.

q1-paper-1h-june-2019-aqa-gcse-maths

Circle the value of y.

11 7.5 9 4

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9
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1 mark

Which of these is not used to prove that triangles are congruent?

Circle your answer.

SSS SAS AAA RHS

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10a
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2 marks

q3-4ma1-2h-qp-jan22-paper2-igcse-maths-1

Triangle A B C is similar to triangle P Q R

space A B equals 4 space cm     P Q equals 12 space cm       R Q equals 16.5 space cm   space A C equals x space cm    P R space equals y space cm

Calculate the length of B C.

10b
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1 mark

Write down an expression for y in terms of x

y = ...................................... 

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11a
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2 marks

A B C and D E F are similar triangles.

q4-4ma1-2h-qp-nov21-paper2-igcse-maths

Work out the length of A B.

11b
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2 marks

Given that B C space equals space 54 space cm,
work out the length of E F.

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12
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2 marks

The diagram shows two cylinders, bold A and bold B

q6-paper2h-june2018-edexcel-igcse-maths

Cylinder bold A has height 1.6 m and radius 0.56 m.

Cylinder bold B is mathematically similar to cylinder bold A.
The height of cylinder bold B is 0.6 m.
Work out the radius of cylinder bold B.

....................................................... m 

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13a
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2 marks

A B C spaceand space D E F spaceare similar triangles.

q6-paper2h-june2019-edexcel-igcse-maths

Work out the length of space D F.

.......................cm

13b
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2 marks

Work out the length of B C.

....................cm

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14
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2 marks

A B C spaceand space D E F spaceare similar triangles.

q4-paper2hr-june2019-edexcel-igcse-maths

Work out the length of D E.

........................cm

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15
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1 mark

Are these two triangles definitely congruent?
Give a reason.

q11a-paper4-nov2019-ocr-gcse-maths

..................... because ..............................................................................................................

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1a
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3 marks

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

Prove that triangle ADP is congruent to triangle CBQ.

1b
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2 marks

Explain why AQ is parallel to PC.

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2
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3 marks

A B C D is a rhombus.

q2-medium-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

M and N are points on B D such that D N space equals space M B.

Prove that triangle D N C is congruent to triangle B M C.

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3
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3 marks

q3-medium-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

P Q space equals space P R.
S is the midpoint of P Q.
T is the midpoint of P R.

Prove triangle Q T R is congruent to triangle R S Q.

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4
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3 marks

q4-medium-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

A B C and A E D are straight lines.
B E and C D are parallel.
B E space equals space 1.5 space cm. space
C D space equals space 6 space cm. space
A D space equals space 5 space cm.

Calculate the length of E D.

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5a
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2 marks

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ABC and EDC are straight lines.
EA is parallel to DB.

EC = 8.1 cm.
DC = 5.4 cm.
DB = 2.6 cm,

Work out the length of AE.

5b
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2 marks

AC = 6.15 cm.

Work out the length of AB.

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6
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3 marks

A B C D is a parallelogram.

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E is the point where the diagonals A C and B D meet.

Prove that triangle A B E is congruent to triangle C D E.

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7
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1 mark

q2-paper1h-june2017-aqa-gcse-maths

Circle the reason why these triangles are congruent.

SSS SAS ASA RHS

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8
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1 mark

Here are four triangles.

q11-paper-1h-nov-2018-aqa-gcse-maths

Which two triangles are congruent?
Circle two letters below.

A B C D

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9
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1 mark

P R T and Q R S are similar triangles.

q22-paper1h-june2017-aqa-gcse-maths

Which of these is equivalent to fraction numerator Q R over denominator P R end fraction?

Circle your answer.

fraction numerator R S over denominator S T end fraction fraction numerator Q S over denominator P T end fraction
fraction numerator P T over denominator Q S end fraction fraction numerator R T over denominator R S end fraction

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10
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3 marks

A comma space B spaceand C are points on a circle.

  • B C bisects angle A B Q.
  • P B Q is a tangent to the circle.

q21-paper1h-specimen2015-aqa-gcse-maths

Angle C B Q space equals space x

Prove that A C space equals space B C

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11
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3 marks

The diagram shows two water towers in Kuwait.

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The real height of tower bold A is 187 straight m.
The real height of tower bold B is 147 straight m.

Ahmed makes a scale model of both towers.

The height of tower bold A on the scale model is 90 cm.

Work out the height of tower bold B on the scale model.
Give your answer correct to the nearest centimetre.

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12a
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2 marks

The diagram shows triangle ABC.
CD is parallel to AB.
A, C and E lie in a straight line.
Angles of size space a degree comma space b degreeand c degree are shown.

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Insert  a degree comma space b degree° or c degree to make this statement true.
Give a reason for your answer.

Angle DCE = ......... because ....................................................................................................

12b
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3 marks

Use the diagram and the answer to part (a) to show that the angles of a triangle add up to 180°.

Give a reason for each statement you make.

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13
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3 marks

In the diagram AB is parallel to CD.
AED and BEC are straight lines.

q14-paper5-june2019-ocr-gcse-maths

Prove that triangle ABE is similar to triangle CDE.

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14
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4 marks

The diagram below shows two triangles.

q11-paper6-june2018-ocr-gcse-maths

Prove that triangle ABC is congruent to triangle ACD.

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15a
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3 marks

Anna estimates the height of a tree.

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Anna holds a ruler vertically so the height of the tree is exactly covered by the ruler.
She is 20 metres from the tree.
The ruler is 30cm long.
The horizontal distance from her eyes to the ruler is 60 cm.

Calculate an estimate of the height of the tree.

.......................... m

15b
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2 marks

Give two reasons why this method may not be suitable to estimate the height of a very tall building.

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16
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3 marks

ABCD is a parallelogram.

q10-paper4-spec2015-ocr-gcse-maths

Prove that triangle ABD is congruent to triangle CDB.

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1
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3 marks

q1-hard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

A O C and B O D are diameters of a circle, centre O.

Prove that triangle A B D and triangle D C A are congruent.

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2
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3 marks

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

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A E C and B E D are straight lines.
Prove that triangle A B E and triangle D C E are similar.
You must give reasons for each stage of your working.

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3
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3 marks

Steve has a photo and a rectangular piece of card.

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The photo is 16 cm by 10 cm.
The card is 30 cm by 15 cm.

Steve cuts the card along the dotted line shown in the diagram below.

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Steve throws away the piece of card that is 15 cm by x cm.
The piece of card he has left is mathematically similar to the photo.

Work out the value of x.

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4
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3 marks

Rectangle A B C D is mathematically similar to rectangle D A E F.

q4-hard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

A B space equals space 10 space cm. space
A D space equals space 4 space cm.

Work out the area of rectangle D A E F.

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5a
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2 marks

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PQR and PTS are straight lines.
Angle PTQ = Angle PSR = 90o.
QT = 4 cm
RS = 12 cm
TS = 10 cm

Work out the area of the trapezium QRST.

5b
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3 marks

Work out the length of PT.

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6
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3 marks

q6-hard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

The diagram shows a triangle A B C.

L M N B is a parallelogram where
    space L is the midpoint of A B,
   M  is the midpoint of A C,
and   N is the midpoint of B C.

Prove that triangle A L M and triangle M N C are congruent.
You must give reasons for each stage of your proof.

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7
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3 marks

The diagram shows a triangle and a trapezium.

q19-paper1h-nov2017-aqa-gcse-maths

Prove that a space equals space b.

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8
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4 marks

In the diagram, AB and DC are parallel lines of equal length.

Prove that angle DAB = angle BCD.

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9
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4 marks

ABCD is a quadrilateral.
AD = AB and CD = CB.

q10-paper5-nov2017-ocr-gcse-maths

Prove that angle ADC is equal to angle ABC.

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1
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4 marks

q1-veryhard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

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

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

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2
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4 marks

A comma space B and C are points on the circumference of a circle centre O.

q2-veryhard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

Prove that angle B O C is twice the size of angle B A C.

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3
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5 marks

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P Q R S T is a regular pentagon.
R comma space U and T are points on a circle, centre O.
Q R and P T are tangents to the circle.
R S U is a straight line.

Prove that S T space equals U T.

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4
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5 marks

The two triangles in the diagram are similar.

q4-veryhard-4-8-congurence-similarity-and-geometry-edexcel-gcse-maths

There are two possible values of x.

Work out each of these values.
State any assumptions you make in your working.

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5
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5 marks

The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm.

The strut is one third of the way up the ladder.

The symmetrical cross section of the ladder shows two similar triangles.

q23-paper-3h-nov-2018-aqa-gcse-maths

Work out the vertical height, h cm, of the ladder.

...........................cm

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6
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4 marks

A and B are points on the circumference of a circle, centre O.
CA and CB are tangents to the circle.

q13-paper6-june2017-ocr-gcse-maths

Prove that triangle OAC is congruent to triangle OBC.

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7
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5 marks

cie-igcse-2020-may-june-p2-tz1-q13

C lies on a circle with diameter A D.
B lies on A C and E lies on A D such that B E is parallel to C D.
A B = 21 cm, C D = 18 cm and B E = 13.5 cm.

Work out the radius of the circle.


............................................ cm

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8a
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3 marks

cie-igcse-2020-oct-nov-p4-tz2-q8c

 The diagonals of the cyclic quadrilateral ABCD intersect at X.  

Explain why triangle ADX is similar to triangle BCX.
Give a reason for each statement you make.

8b
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2 marks

AD  = 10 cm, BC  = 8 cm, BX  = 5 cm, CX  = 7 cm.  

Calculate DX.

 

DX = ........................................... cm

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9
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3 marks

cie-igcse-2020-may-jun-p4-tz3-q8c

A, B, C and D are points on the circle, centre O.
M is the midpoint of AB and N is the midpoint of CD.
OM = ON
 
Explain, giving reasons, why triangle OAB is congruent to triangle OCD.

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10a
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2 marks

cie-igcse-2018-oct-nov-p4-tz2-q10b

A solid metal cone has radius 10 cm and height 36 cm.

Calculate the volume of this cone.
[The volume, V, of a cone with radius r and height h is  V equals 1 third straight pi r squared h.]
 

 

......................................... cm3

10b
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4 marks

The cone is cut, parallel to its base, to give a smaller cone.

cie-igcse-2018-oct-nov-p4-tz2-q10bii

The volume of the smaller cone is half the volume of the original cone.

The smaller cone is melted down to make two different spheres.

The ratio of the radii of these two spheres is 1 : 2.


Calculate the radius of the smaller sphere.

[The volume, V, of a sphere with radius r is  V equals 4 over 3 straight pi r cubed.]

.......................................... cm

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