KS3 Maths · Year 7 · Chapter 5: Congruent shapes · Lesson 2

Congruent triangles

Year 7About 60 minutesNo calculator46 marks

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Date

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Quick questions to start.

A1
Needed today
Two angles of a triangle are $54°$ and $77°$ . Work out the third angle.
[1 mark]
Answer
A2
Needed today
What is the name of the longest side of a right-angled triangle?
[1 mark]
Answer
A3
Last lesson
Are a $3$ cm by $8$ cm rectangle and a $4$ cm by $6$ cm rectangle congruent? Answer yes or no.
[1 mark]
Answer
A4
From chapter 4
Increase $£60$ by $15%$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it. All lengths are in cm.

Example 1

Are triangles $A B C$ and $P QR$ congruent? Give the condition.

Not drawn accurately

Hint The four conditions: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and a side), RHS (right angle, hypotenuse, side).

B1
Show that triangles $D E F$ and $L M N$ are congruent. Give the condition. Which side of $L M N$ is equal to $D F$ ?
[3 marks]
Not drawn accurately
Answer

Example 2

Show that triangles $A B C$ and $X Y Z$ are congruent.

Not drawn accurately

Hint Work out the missing angles first, using the angle sum of a triangle.

B2
Show that triangles $G H I$ and $K L M$ are congruent. Give the condition.
[3 marks]
Not drawn accurately
Answer

Example 3

Are triangles $A B C$ and $P QR$ congruent? Give the condition.

Not drawn accurately

Hint Both have a right angle. Look for the hypotenuse.

B3
Are triangles $D E F$ and $S T U$ congruent? Give the condition. Which angle of $S T U$ is equal to angle $E$ ?
[3 marks]
Not drawn accurately
Answer

Practice

Year 7

C1
Two triangles both have angles $40°$ , $65°$ and $75°$ . Must they be congruent? Give a reason.
[2 marks]
Answer
C2
Sam draws a triangle with sides $8$ cm and $7$ cm, and an angle of $50°$ opposite the $7$ cm side. He says, "Everyone who draws this will get the same triangle." Is he right? Give a reason.
[2 marks]
Answer

C3
Triangle $T U V$ has $T U = 10$ cm, angle $U = 81°$ and angle $V = 61°$ . Work out angle $T$ . Show that triangles $J K L$ and $T U V$ are congruent. Write down the length of $U V$ .
[3 marks]
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Answer
C4
$A B C D$ is a rectangle. Its diagonals cross at $O$ . How many other triangles in the diagram are congruent to triangle $A B C$ ?
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Amy says, "Triangles $A B C$ and $D E F$ both have angles of $50°$ and $70°$ and a side of $6$ cm. So they are congruent."
Is Amy correct? Give a reason for your answer.
[3 marks]
Not drawn accurately

D2
Triangles $A B C$ and $P QR$ are congruent. $A$ matches $P$ , $B$ matches $Q$ and $C$ matches $R$ .
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(a)
Work out the value of $x$ .
[2 marks]
Answer
(b)
Work out the length of $B C$ .
[2 marks]
Answer
(c)
$C A = 11$ cm. Work out the perimeter of triangle $P QR$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
$A B C$ is an isosceles triangle with $A B = A C$ . The line $A D$ cuts angle $B A C$ in half. $D$ is on $B C$ .
(a)
Show that triangles $A B D$ and $A C D$ are congruent.
[3 marks]
(b)
$B C = 13.6$ cm. Work out the length of $B D$ .
[1 mark]
Answer
(c)
Explain why angle $A D B = 90°$ .
[2 marks]
Total for D3: 6 marks
D4
Here are four triangles. Which two triangles are congruent? Give the condition. Give a reason why each of the other two is not congruent to them.
[4 marks]
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Extension

Stretch

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

E1
$A B C D E F$ is a regular hexagon. Triangle $A B D$ is shaded. How many triangles with their three corners at corners of the hexagon are congruent to triangle $A B D$ ? Count triangle $A B D$ itself.
[5 marks]
Hint 1 · What to try
$A D$ goes through the centre of the hexagon. Every triangle congruent to $A B D$ has one side that goes through the centre.
Hint 2 · The first line
There are $3$ lines through the centre, $A D$ , $B E$ and $C F$ . How many triangles like $A B D$ use the line $A D$ ?
Hint 3 · The full method
$A D$ makes $4$ triangles: $A B D$ , $A C D$ , $A D E$ and $A D F$ . The same happens for $B E$ and $C F$ . That is $3 \times 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use SSS, SAS, ASA and RHS to show that two triangles are congruent.
I can use congruent triangles to find a missing side or angle.
I know why three equal angles are not enough.
I can find congruent triangles in a shape.

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