GCSE Maths · Higher · Chapter 7: Transformations, constructions and loci · Lesson 1

Congruent triangles

Grades 4–8About 50 minutesNo calculator36 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-congruent-triangles/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Angles on a straight line are $5 x$ and $x$ . Find $x$ .
[1 mark]
Answer
A2
From chapter 2
Work out $\frac{3}{4} \div \frac{1}{2}$ .
[1 mark]
Answer
A3
From chapter 5
A train travels at 90 km/h for 20 minutes. How far does it go?
[1 mark]
Answer
A4
Last lesson
The bearing of B from A is $06 0^{\circ}$ . Find the bearing of A from B.
[1 mark]
Answer

Example, then your turn

Grades 4–7

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

Example 1

In triangles $A B C$ and $P QR$ , $A B = P Q = 7$ cm, $A C = P R = 11$ cm and angle $A =$ angle $P = 4 8^{\circ}$ . Name the condition that proves them congruent. $B C$ measures $8.2$ cm and angle $B$ measures $9 3^{\circ}$ . Write down $QR$ and angle $Q$ .

Hint Is the equal angle between the two equal sides? Then match the letters in order:

A \to P

B \to Q

C \to R

B1
In triangles $D E F$ and $X Y Z$ , $D E = X Y = 9$ cm, $D F = X Z = 14$ cm and angle $D =$ angle $X = 6 3^{\circ}$ . Name the condition that proves them congruent. $E F$ measures $12.8$ cm and angle $E$ measures $7 8^{\circ}$ . Write down $Y Z$ and angle $Y$ .
[3 marks]
Answer $\dots\dots$ ; $Y Z = \dots\dots$ cm; $Y = \dots \dots^{\circ}$

Example 2

$A B$ is parallel to $D C$ . $A C$ and $B D$ meet at $M$ , and $A M = M C$ . Prove that triangle $A B M$ is congruent to triangle $C D M$ .

Hint Find two pairs of equal angles and one pair of equal sides, and give a reason for each.

B2
$P Q$ is parallel to $S R$ . $P R$ and $QS$ meet at $X$ , and $QX = X S$ . Prove that triangle $P QX$ is congruent to triangle $R S X$ .
[3 marks]

Practice

Grades 5–7

C1
Name the condition that proves these triangles congruent. Which side of triangle $P QR$ is equal to $B C$ ?
[2 marks]
Answer $\dots\dots\dots$ and $\dots\dots$
C2
Are triangles $X Y Z$ and $L M N$ congruent? Give a reason. Which side of $L M N$ is equal to $X Z$ ? Work out angle $N$ .
[3 marks]
Not drawn accurately
Answer $\dots\dots$ , $\dots\dots$ ; $\dots\dots$ ; $N = \dots \dots^{\circ}$

C3
Triangle $A B C$ is congruent to triangle $P QR$ , with the letters in order. Find $x$ and $y$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ $y = \dots\dots$
C4
$A B = D E$ and angle $B$ = angle $E$ . Write down one more fact that would prove the triangles congruent by SAS, and another that would prove it by ASA.
[2 marks]
AnswerSAS: $\dots\dots\dots\dots$ ASA: $\dots\dots\dots\dots$

Exam-style questions

Grades 6–8

D1
Triangle $A B C$ has $A B = A C = 15$ cm and $B C = 18$ cm. $A D$ is perpendicular to $B C$ .
Not drawn accurately
(a)
Prove that triangle $A B D$ is congruent to triangle $A C D$ .
[2 marks]
(b)
Hence work out the area of triangle $A B C$ . You must show your working.
[4 marks]
Answer
Total for D1: 6 marks

D2
Here are four triangles, $T$ , $X$ , $Y$ and $Z$ . Ali says, " $X$ , $Y$ and $Z$ are all congruent to $T$ , because each of them has a $9$ cm side and a $7$ cm side."
Ali is wrong.
(a)
Which of $X$ , $Y$ and $Z$ must be congruent to triangle $T$ ? Give the condition.
[2 marks]
AnswerTriangle $\dots\dots$ by $\dots\dots\dots$
(b)
Explain why triangle $X$ need not be congruent to triangle $T$ .
[2 marks]
(c)
Lena says, "A triangle with the same three angles as triangle $T$ must be congruent to triangle $T$ ." Explain why Lena is wrong.
[2 marks]
Total for D2: 6 marks

Extension

Grade 8 and beyond

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

E1
$A B C D$ and $A E F G$ are squares, both lettered anticlockwise, with the corner $A$ in common. Prove that $B E = D G$ .
[4 marks]
Hint 1 · What to try
Look for one triangle holding $B E$ and one holding $D G$ . Both have a corner at $A$ .
Hint 2 · The first line
Compare triangle $A B E$ with triangle $A D G$ . Which sides are sides of the squares?
Hint 3 · The full method
$A B = A D$ and $A E = A G$ . Angle $B A E = 9 0^{\circ} +$ angle $D A E$ = angle $D A G$ . SAS gives $B E = D G$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can say which of SSS, SAS, ASA and RHS proves two triangles congruent, and match their letters.
I can write a congruence proof with a reason for every step.
I can explain why three angles, or two sides and an angle outside them, prove nothing.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers