KS3 Maths · Year 7 · Chapter 5: Congruent shapes · Lesson 1

Congruent shapes

Year 7About 60 minutesNo calculator46 marks

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Quick questions to start.

A1
Needed today
Three angles of a quadrilateral are $85°$ , $95°$ and $110°$ . Work out the fourth angle.
[1 mark]
Answer
A2
Needed today
How many lines of symmetry does a rectangle that is not a square have?
[1 mark]
Answer
A3
Last lesson
A price falls from $£50$ to $£40$ . Work out the percentage decrease.
[1 mark]
Answer
A4
From chapter 3
The probability that a spinner lands on $4$ is $0.35$ . Work out the probability that it does not land on $4$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Which two of the shapes $A$ , $B$ , $C$ and $D$ are congruent?

Hint Congruent shapes have the same shape and the same size. A shape can be turned or flipped over and still be congruent.

B1
Which two of the shapes $E$ , $F$ , $G$ and $H$ are congruent? Give a reason.
[2 marks]
Answer

Example 2

Which triangle is the odd one out? Give a reason.

Hint Look at the two shorter sides of each right-angled triangle.

B2
Which shape is the odd one out? Give a reason.
[2 marks]
Answer

Example 3

$A B C D$ and $P QR S$ are congruent ( $A$ matches $P$ , $B$ matches $Q$ , $C$ matches $R$ , $D$ matches $S$ ). Write down the length of $QR$ and the size of angle $S$ .

Not drawn accurately

Hint Matching sides are equal and matching angles are equal. Use the order of the letters.

B3
$E F G H$ and $K L M N$ are congruent ( $E$ matches $K$ , $F$ matches $L$ , $G$ matches $M$ , $H$ matches $N$ ). $E F = 8$ cm, angle $E = 95°$ , angle $F = 70°$ , angle $G = 110°$ . Write down the length of $K L$ . Work out angle $N$ .
[3 marks]
Not drawn accurately
Answer

Practice

Year 7

C1
Which of the shapes $B$ , $C$ and $D$ are congruent to shape $A$ ?
[2 marks]
Answer
C3
Shape $Q$ is an enlargement of shape $P$ with scale factor $3$ . Are $P$ and $Q$ congruent? Give a reason.
[2 marks]
Answer

C4
On the diagram, draw lines to split triangle $A B C$ into four congruent triangles.
[2 marks]
C5
Which of these pairs must be congruent? (i) two circles that each have radius $5$ cm (ii) two squares that each have perimeter $20$ cm (iii) two rectangles that each have perimeter $20$ cm
[3 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Kai says, "Shapes $P$ and $Q$ are not congruent, because $Q$ is upside down."
Is Kai correct? Give a reason for your answer.
[2 marks]

D2
The diagram shows a right-angled triangle. Two of these triangles are joined so that two equal sides match exactly.
(a)
The two triangles are joined along their $5$ cm sides. One shape that can be made is a rectangle. Write down the name of the other shape that can be made.
[1 mark]
Answer
(b)
Work out the perimeter of the rectangle.
[2 marks]
Answer
(c)
The two triangles are now joined along their $4$ cm sides to make an isosceles triangle. Work out the perimeter of the isosceles triangle.
[2 marks]
Answer
Total for D2: 5 marks
D3
Pentagons $A B C D E$ and $P QR S T$ are congruent. $A$ matches $P$ , $B$ matches $Q$ , $C$ matches $R$ , $D$ matches $S$ and $E$ matches $T$ . Work out the perimeter of pentagon $A B C D E$ . You must show your working.
[4 marks]
Not drawn accurately
Answer
D4
The large rectangle in the diagram is made from three congruent rectangles. The perimeter of the large rectangle is $60$ cm. Work out the area of one of the small rectangles. You must show your working.
[5 marks]
Answer
D5
$A B C D E F$ is a regular hexagon with perimeter $48$ cm. It is split into six congruent equilateral triangles.
(a)
Work out the perimeter of one of the triangles.
[2 marks]
Answer
(b)
The hexagon is cut along $A D$ into two congruent pieces. Work out the perimeter of one piece.
[3 marks]
Answer
Total for D5: 5 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Shapes are made from $5$ equal squares joined edge to edge. Two shapes count as the same if they are congruent, like the two shapes in the diagram. How many different shapes can be made?
[5 marks]
Hint 1 · What to try
Start with all $5$ squares in a line. Then try $4$ in a line and move the fifth square, then $3$ in a line, and so on.
Hint 2 · The first line
With $4$ in a line there are only two different shapes. Flip and turn each new shape to check it is not one you already have.
Hint 3 · The full method
Sort by the longest line of squares: a line of $5$ gives $1$ shape, $4$ gives $2$ , $3$ gives $8$ and $2$ gives $1$ . That is $1 + 2 + 8 + 1 = 12$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can decide whether two shapes are congruent.
I know a reflected or rotated shape is congruent to the original.
I can find matching sides and angles in congruent shapes.
I can split a shape into congruent pieces.

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