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Grade 4 · Foundation and Higher · free GCSE maths worksheet

Congruent shapes

Spotting congruent triangles and shapes.

11 original questions, 26 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Here are the side lengths of four triangles.

Triangle A: 5 cm, 7 cm, 9 cm
Triangle B: 5 cm, 7 cm, 8 cm
Triangle C: 9 cm, 7 cm, 5 cm
Triangle D: 5 cm, 6 cm, 9 cm

Write down the letters of the two triangles that are congruent to each other.

1 mark
Show answer
A and C
2

In triangles PQR and XYZ,

PQ = XY = 6 cm
angle PQR = angle XYZ = 40°
QR = YZ = 8 cm

The two triangles are congruent.

Write down the congruence condition that shows the triangles are congruent.

1 mark
Show answer
SAS
3

Triangle ABC has AB = 5 cm, BC = 7 cm and AC = 9 cm.
Triangle PQR has PQ = 5 cm, QR = 7 cm and PR = 9 cm.

Are the two triangles congruent? Give a reason for your answer.

2 marks
Show answer
Yes, all three sides equal (SSS)
4

Here are the measurements of four rectangles.

  • Rectangle A: 6 cm by 4 cm
  • Rectangle B: 4 cm by 6 cm (turned through a right angle)
  • Rectangle C: 6 cm by 5 cm
  • Rectangle D: 5 cm by 6 cm

Write down the two pairs of rectangles that are congruent.

2 marks
Show answer
A and B; C and D
5

Triangles ABC and PQR are congruent. A matches P, B matches Q and C matches R.

AB = 7 cm, BC = 9 cm, PR = 11 cm and angle ABC = 64°.

  1. Write down the length of QR.
  2. Write down the length of AC.
  3. Write down the size of angle PQR.
3 marks
Show answer
a) 9 cm b) 11 cm c) 64°
6

For each pair of triangles, write down the condition (SSS, SAS, ASA, RHS) that shows they are congruent, or write "not enough information".

  1. Both triangles have sides of 6 cm and 8 cm with an angle of 50° between them.
  2. Both triangles have a side of 7 cm with an angle of 40° at one end and an angle of 60° at the other end.
  3. Both triangles have a right angle, a hypotenuse of 13 cm and one other side of 5 cm.
  4. Both triangles have angles of 50°, 60° and 70°.
4 marks
Show answer
a) SAS b) ASA c) RHS d) not enough information
7

In triangle ABC, AB = 8 cm, BC = 6 cm and angle BAC = 40°.
In triangle DEF, DE = 8 cm, EF = 6 cm and angle EDF = 40°.

Ahmed says "The triangles are congruent because two sides and an angle are equal (SAS)."

Explain why Ahmed may not be correct.

2 marks
Show answer
The angle is not between the two sides, so SAS does not apply
8

The diagram shows five quadrilaterals, labelled A, B, C, D and E, drawn on a centimetre grid.

  1. Write down the letters of the pair of congruent quadrilaterals.
  2. Write down the mathematical name of quadrilateral A.
ABCDE
2 marks
Show answer
a) A and C b) parallelogram
9

ABCD is a quadrilateral with AB = CD and BC = DA. The diagonal AC is drawn.

Prove that triangle ABC is congruent to triangle CDA.

3 marks
Show answer
AB = CD, BC = DA, AC common: SSS
10

The straight lines AB and CD cross at P. P is the midpoint of AB and P is the midpoint of CD.

Prove that triangle APC is congruent to triangle BPD.

3 marks
Show answer
AP = PB, CP = PD, vertically opposite angles equal: SAS
11

A and B are points on a circle with centre O. M is the midpoint of the chord AB. OA, OB and OM are drawn.

Prove that triangle OAM is congruent to triangle OBM.

3 marks
Show answer
OA = OB (radii), AM = MB, OM common: SSS

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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