KS3 Maths · Year 7 · Chapter 5: Congruent shapes · Lesson 3

Using congruent triangles to solve problems

Year 7About 60 minutesNo calculator48 marks

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Date

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Quick questions to start.

A1
Needed today
Two straight lines cross. One of the angles is $38°$ . Work out the size of an angle next to it.
[1 mark]
Answer
A2
Needed today
An isosceles triangle has an angle of $40°$ between its two equal sides. Work out one of the other angles.
[1 mark]
Answer
A3
Last lesson
Two triangles have two pairs of equal sides and the angles between them are equal. Which condition shows they are congruent?
[1 mark]
Answer
A4
From chapter 4
A price goes up from $£80$ to $£92$ . Work out the percentage increase.
[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

$P QR S$ is a kite with $P Q = P S$ and $QR = S R$ . Show that angle $P QR =$ angle $P S R$ .

Not drawn accurately

Hint Give a reason for each pair of equal parts. A side that is in both triangles is a common side.

B1
$W X Y Z$ has $W X = Y Z$ and $X Y = Z W$ . Show that angle $W X Y =$ angle $Y Z W$ .
[3 marks]
Not drawn accurately

Example 2

$A B$ and $C D$ cross at $X$ . $X$ is the midpoint of $A B$ and of $C D$ . $A C = 7$ cm. Show that triangles $A X C$ and $B X D$ are congruent. Work out $B D$ .

Not drawn accurately

Hint Where two straight lines cross, the opposite angles are equal.

B2
$J K$ and $L M$ cross at $N$ , the midpoint of both. $J L = 9$ cm and angle $L J N = 43°$ . Show that triangles $J N L$ and $K N M$ are congruent. Write down the length of $K M$ and the size of angle $M K N$ .
[3 marks]
Not drawn accurately
Answer

Example 3

$A B C$ is isosceles with $A B = A C$ . $A D$ is perpendicular to $B C$ . Angle $B A D = 24°$ . Show that triangles $A B D$ and $A C D$ are congruent. Work out angle $A C D$ .

Not drawn accurately

Hint Both triangles have a right angle at

D

. Find the hypotenuse.

B3
$O$ is the centre of a circle. $A B$ is a chord of length $14$ cm. $O M$ is perpendicular to $A B$ . Show that triangles $O A M$ and $O B M$ are congruent. Work out the length of $A M$ .
[3 marks]
Not drawn accurately
Answer

Practice

Year 7

C1
$A B C D$ is a parallelogram. Show that triangles $A B D$ and $C D B$ are congruent. Give a reason for each pair of equal parts.
[3 marks]
Not drawn accurately
C2
In the kite $A B C D$ , $A B = C B$ and $A D = C D$ . Angle $A D B = (3 x + 10) °$ and angle $C D B = (5 x - 14) °$ . Explain why the two angles are equal. Work out the value of $x$ and the size of angle $A D B$ .
[3 marks]
Not drawn accurately
Answer

C3
$A$ , $B$ and $C$ lie on a straight line with $A B = B C$ . $D A = D C$ . Show that angle $D B A = 90°$ .
[3 marks]
Not drawn accurately
C4
$A B C D$ is a square. $E$ is the midpoint of $A B$ . Show that $D E = C E$ .
[3 marks]

Exam-style questions

GCSE Foundation

D1
$P Q$ is parallel to $R S$ . $P S$ and $QR$ cross at $X$ . Ben says, "Angle $QP X =$ angle $R S X$ and angle $P QX =$ angle $S R X$ (alternate angles), and angle $P X Q =$ angle $S X R$ (vertically opposite). So triangles $P QX$ and $S R X$ are congruent."
Is Ben correct? Give a reason for your answer.
[3 marks]
Not drawn accurately

D2
$A C$ is parallel to $D B$ . $A B$ and $C D$ cross at $E$ , and $A E = E B$ .
Not drawn accurately
(a)
Show that triangles $A E C$ and $B E D$ are congruent. Give a reason for each pair of equal parts.
[3 marks]
(b)
$A C = 6.4$ cm and $C E = 5.1$ cm. Work out the lengths of $B D$ and $C D$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
$M$ is the midpoint of $A B$ . The point $P$ is on the line through $M$ that is perpendicular to $A B$ .
(a)
Show that triangles $P A M$ and $P B M$ are congruent.
[3 marks]
(b)
$A B = 10$ cm. The perimeter of triangle $P A B$ is $36$ cm. Work out the length of $P A$ .
[2 marks]
Answer
Total for D3: 5 marks
D4
$A B C$ is an isosceles triangle with $A B = A C$ . $D$ and $E$ are on $B C$ with $B D = C E$ .
Not drawn accurately
(a)
Show that triangles $A B D$ and $A C E$ are congruent.
[3 marks]
(b)
Angle $A B C = 52°$ and angle $B A D = 18°$ . Work out the size of angle $A E C$ .
[2 marks]
Answer
Total for D4: 5 marks

Extension

Stretch

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

E1
$A B C D$ is a square. $E$ is the midpoint of $B C$ and $F$ is the midpoint of $C D$ . The lines $A E$ and $B F$ cross at $P$ . Prove that angle $A P B = 90°$ .
[5 marks]
Hint 1 · What to try
Look at triangles $A B E$ and $B C F$ . Are they congruent?
Hint 2 · The first line
They are congruent by SAS, so angle $B A E =$ angle $C B F$ . Call this angle $a$ .
Hint 3 · The full method
Angle $A B P = 90 - a$ . Now use the angles in triangle $A B P$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove two triangles are congruent, with a reason for each pair of equal parts.
I can use common sides, vertically opposite angles and alternate angles in a proof.
I can use congruent triangles to find a missing side or angle.
I know that three equal angles do not prove two triangles are congruent.

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