Congruence, Similarity & Geometrical Proof (OCR GCSE Maths: Higher): Exam Questions

3 hours49 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.

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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

Are these two triangles definitely congruent?
Give a reason.

q11a-paper4-nov2019-ocr-gcse-maths

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

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6a
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2 marks
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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

6b
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1 mark

Write down an expression for y in terms of x

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

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7a
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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.

7b
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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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8
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2 marks

The diagram shows two cylinders, bold A and bold B

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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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9a
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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

9b
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2 marks

Work out the length of B C.

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

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10
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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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11
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1 mark
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Circle the reason why these triangles are congruent.

ASA        

 RHS  

SAS  

 SSS

       

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12
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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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13
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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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14
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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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1a
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3 marks
screenshot-2023-05-26-at-4-56-56-pm

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.

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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
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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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7a
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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 ....................................................................................................

7b
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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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8
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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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9
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4 marks

The diagram below shows two triangles.

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Prove that triangle ABC is congruent to triangle ACD.

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10a
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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

10b
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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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