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29 Congruent triangles
GCSE Mathematics for OCR (Foundation)
In this chapter you will learn how to …
• prove that two triangles are congruent using the cases SSS, ASA, SAS, RHS.
• apply congruency in calculations and simple proofs.
or more resources relating
F
to this chapter, visit GCSE
Mathematics Online.
Using mathematics: real-life applications
Congruent triangles are used in construction to reinforce structures that need to be strong and stable.
“When designing any bridge I have to allow for
reinforcement. This ensures that the bridge doesn’t
collapse under heavy traffic. Any bridge I design has
(Structural engineer)
many congruent triangles.”
Before you start …
Ch 5
You need to know
how to label angles
and shapes that
are equal.
C
1 Here are two identical triangles.
D
a Write down a pair of sides
A
that are equal in length.
b What angle is equal in size
B
to angle BAC?
c Write down another pair of
F
angles that are equal in size.
Ch 9
Ch 5,
9
Ch 5
You need to know
basic angle facts.
2 Match up the correct
E
A
statement with the correct diagram.
a Vertically opposite angles
are equal.
b Alternate angles are equal.
c Corresponding angles are equal.
C
B
You should be
able to apply angle
facts to find angles
in figures and to
justify results in
simple proofs.
3 Decide whether each statement is true
You need to
know and be
able to apply
the properties
of triangles and
quadrilaterals.
4 What is the value of x? Choose the correct answer.
A
B
or false.
a Angle DBE 5 40° (alternate to angle ADB).
bAngle BEC 5 50° (complementary to
40°
angle ADB).
c Angle BDE 5 angle BED 5 70°.
D
dTriangle ABD, triangle BDE and triangle BCE are congruent.
E
A
D
x
A60°
B30°
C 45° P
D50°
B
450
C
C
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29 Congruent triangles
Assess your starting point using the Launchpad
Step 1
1 Identify which pairs of triangles are congruent. Give reasons for
your decisions.
a
Go to
Section 1:
Congruent triangles
6 cm
8 cm
8 cm
6 cm
b
9 mm
4 mm
6 mm
4 mm
6 mm
9 mm
c
2m
25°
80°
25°
75°
2m
✓
Step 2
2 Quadrilateral ABCD is a kite.
A
a Prove that triangle ACD is congruent
to triangle ACB.
b Prove that angle ADC 5 angle ABC
D
M
B
Go to
Section 2:
Applying congruency
C
✓
Go to
Chapter review
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
451
GCSE Mathematics for OCR (Foundation)
Key vocabulary
congruent: shapes that are
identical in shape and size.
Section 1: Congruent triangles
Congruent triangles are identical in shape and all corresponding
measurements are equal.
The corresponding sides are equal in length. The corresponding angles are
the same size.
x
x
A
B
y
Tip
If you place two congruent
triangles on top of each other the
angles and sides will match up.
The matching sides and angles
are the corresponding sides or
angles.
y
z
z
Congruent triangles can have different orientations. When the triangles are
in different orientations you need to think carefully about the corresponding
sides and angles.
This diagram shows triangle A and B from the example above in different
orientations.
x
x
A
B
y
z
y
z
Two triangles are congruent if one of the following sets of conditions is true.
Side side Side or SSS: the
three sides of one triangle
are equal in length to the
three sides of the other
triangle.
Angle side angle or ASA:
two angles and one side of
one triangle are equal to the
corresponding two angles
and side of another triangle.
D
A
B
C
F
E
F
D
A
C
B
E
Key vocabulary
included angle: the angle
between two lines that meet at a
vertex.
Side angle side or SAS:
two sides and the included
angle of one triangle are
equal to two sides and the
included angle of the other
triangle.
Right angle hypotenuse
side or RHS: the
hypotenuse and one side of
a right-angled triangle are
equal to the hypotenuse and
one other side of the other
right-angled triangle.
452
D
A
F
B
C
A
B
E
D
C
F
E
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29 Congruent triangles
Tip
It is important to write the letters of the vertices of the two triangles in the
correct order. When we write that triangle ABC is congruent to triangle DEF, it
means that:
 5 D̂
B
E
B̂ 5 Ê
Ĉ 5 F̂
and
AB 5 DE
A
C
D
F
AC 5 DF and
BC 5 EF.
The conditions in the table are the minimum conditions for proving that
triangles are congruent. No other combinations of side and angle facts are
sufficient to tell us whether a triangle is congruent or not.
For example:
Two triangles with all their angles equal can still be very different sizes.
FF
DD
AA
CC
BB
EE
If you are given two triangles that have two equal sides and one equal angle,
but where the equal angle is not included (between the two given sides), you
do not know if the triangles are congruent or not. The third side might have a
different length in the two triangles.
G
6 cm6 cm
J
G
7 cm7 cm
J
7 cm7 cm
6 cm6 cm
40° 40°
H H
40° 40°
I
I
K
K
L
L
Although a pair of triangles with one of these sets of information could still
be congruent, the conditions given are not sufficient to prove that they are.
Tip
If two congruent shapes are drawn in different orientations it is sometimes hard
to see which angles and sides match each other. To help you, trace one shape
onto tracing paper and label its vertices, then rotate and/or flip the paper over
to help see which sides and angles match up.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
453
GCSE Mathematics for OCR (Foundation)
Work it out 29.1
Here are three proofs for congruence
for the pair of triangles.
P
X
85°
85°
5 cm
Which one uses the correct reasoning?
Why are the others incorrect?
5 cm
30°
65°
R
6 cm
Q
Y
6 cm
Z
Option A
Option B
Option C
In triangle PQR and triangle
XYZ:
In triangle PRQ and triangle
XYZ:
In triangle PRQ and triangle
XYZ:
PR 5 XY 5 5 cm
PR 5 XY 5 5 cm
angle P 5 angle X 5 85°
In triangle PRQ, angle Q 5 65°
(sum of angles in a triangle)
RQ 5 YZ 5 6 cm
angle Q 5 angle Z 5 65°
so triangle PQR is congruent
to triangle XYZ (SAS).
RQ 5 YZ
RQ 5 YZ 5 6 cm
In triangle XYZ, angle Y 5 30°
(sum of angles in a triangle)
so the triangles are congruent. so triangle PRQ is congruent
to triangle XYZ (SAS).
Exercise 29A
1 Match up each of the congruency descriptions (SSS, ASA, SAS, RHS)
with each pair of triangles below:
b
a
A
E
C
F
K
H
J
B
D
L
c
t
s
E
d
s
t
H
E
D
u
u
G
F
2 Which of the following figures show a pair of congruent triangles? In
your answer, state whether the triangles are congruent or not, or whether
there is insufficient information. Write the triangles with the vertices in
the correct order andDgive
the
reasons
as SSS, SAS, ASA or RHS.
J
J 5 cm5 cmL
D3 cm
3 cm
a a
E E b b
G G
A A
a a
3 cm3 cm A
B3 cm
B3 cm
B
B
D D3 cm3 cm
E
A3 cm3 cm
3 cm3 cm
3 cm3 cm
3 cm3 cm
C C
C CF F
F
c c
c c
P
P
M M
M 100°
M
100°
N
10° 100°
10° 100°
P
5 cm5 cm
N
10° 10°
P
5 cm5 cm
454
b b
4 cm4 cmG
4 cm4 cm
H H
H
H
F
R R
Q Q
10°
10°
Q100°
Q100° R R
10° 10°
100°100°
5 cm5 cm
N
5 cm5 cm
N
S S
S
E
S
d dT
d dT
G
L
20°5 cm
20°
J
J 5 cm
L
4 cm4 cm
5 cm5 cm
20° 20°
L
20°5 cm
20°5 cm
4 cm4 cm
I
I
20° 20°
K K
I
I
K K
T
W W
T
20° 20°8 cm8 cm
W W
8 cm8 cm 20° 20°
8 cm8 cm 20° 20°
20° 20°8 cm8 cm
U
U
V
VY
Y
X
X
U
U
V
VY
Y
X
X
© Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms
29 Congruent triangles
3 Prove that triangle ABC is congruent
B
D
to triangle DEC.
C
A
E
4 Write down two different proofs for congruence of triangles DEF
and DGF.
D
E
9m
F
9m
5 In the diagram, PQ is parallel to SR and
G
R
P
QT 5 TR 5 2 cm.
2 cm
Prove that triangle PQT is congruent to
triangle SRT.
T
2 cm
50°
S
Q
6 Prove that triangles ABE and CBD in the figure are congruent, giving full
reasons.
A
B
E
C
D
7 Triangle ABD is isosceles. AC is the perpendicular height. Prove that
triangle ABC is congruent to triangle ADC.
A
B
C
Tip
What do we know about line AC
in an isosceles triangle?
D
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
455
GCSE Mathematics for OCR (Foundation)
8 In the figure below, PR 5 SU and RTUQ is a kite.
Prove that triangle PQR is congruent to triangle SQU.
P
U
T
Q
R
S
9 ABCD in the figure is a kite.
A
D
B
E
C
Prove that:
a triangle ADB is congruent to triangle CDB.
b triangle AED is congruent to triangle CED.
10 Quadrilateral ABCD is a rhombus.
A
D
E
B
C
Prove that:
a triangle AED is congruent to triangle CEB.
b triangle AEB is congruent to triangle CED.
Section 2: Applying congruency
Whenever you learn new skills and concepts in geometry, you add them to
your toolbox and use them in problem-solving. So you will need to combine
what you’ve learnt previously with your new skills to solve problems.
The steps in the following framework are useful for solving geometry
problems.
456
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29 Congruent triangles
Problem-solving framework
In the diagram, AM 5 BM and PM 5 QM.
Q
A
Tip
a Prove that triangle AMP is congruent
to triangle BMQ.
You saw in Chapter 5 that the
symbol // means ‘is parallel to’.
M
b Prove that AP // QB.
B
P
Steps for approaching a
problem‑solving question
What you would do for this example
Step 1: Read the question carefully to
decide what you have to find.
Find mathematical evidence to show that the triangle AMP is
congruent to triangle BMQ; look for SSS, SAS, RHS or ASA. Find
evidence to show that lines AP and QB are parallel.
Step 2: Write down any further
information that might be useful.
Lines AB and PQ intersect, so the two triangles also have vertically
opposite angles. Vertically opposite angles are equal (see Chapter 9 if
you need to).
Step 3: Decide what method you’ll
use.
We are given two equal sides and we can see that the included angle is
also equal, so use SAS to prove congruence.
Step 4: Set out your working clearly.
a In triangles AMP and BMQ:
AM 5 BM (given).
PM 5 QM (given).
angle AMP 5 angle BMQ (vertically opposite angles at M).
So triangle AMP is congruent to triangle BMQ (SAS).
b Angle APM 5 BQM (matching angles of congruent triangles).
So AP // QB (alternate angles are equal).
Worked example 1
Triangle DEF is divided by GF into two smaller triangles.
Prove that FG is perpendicular to DE in the diagram.
E
G
D
In triangle FGD and triangle FGE:
Side FG is common to both triangles.
F
DG 5 EG (given)
DF 5 EF (given)
So the triangles are congruent (SSS).
Angle DGF 5 angle EGF and the two angles lie on a straight line (angles
on a straight line add up to 180°).
So each angle 5 90°, and FG is perpendicular to DE.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
457
GCSE Mathematics for OCR (Foundation)
Exercise 29B
1 In the diagram, prove that KL 5 ML.
J
K
L
M
2 Use the facts given in the
A
D
diagram to:
a prove that angle
E
ABE 5 angle EDC.
b prove that quadrilateral
B
ABCD is a parallelogram.
C
3 In the quadrilateral, SP 5 SR and
P
S
QP // RS. Angle QRP 5 56°.
a Calculate the size of angle PQR
and give reasons.
b Find the size of angle PSR and
56°
give reasons.
Q
R
4 In the diagram, PQ 5 PT , QR 5 ST and angle PQR 5 angle PTS. Prove
that triangle PRS is isosceles.
P
Q
R
S
T
5 In the figure below, prove that:
a triangle AEB is congruent to triangle CEB.
b angle EAD 5 angle ECD.
B
E
A
458
D
C
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29 Congruent triangles
6 In quadrilateral ABCD, AD 5 BC and AD // BC.
Prove that angle ABC 5 angle ADC.
A
D
1
1
B
C
Checklist of learning and understanding
Congruent triangles
You can prove that two triangles are congruent using one of the four cases
of congruence:
○Side side side or SSS:
the three sides of one triangle are equal in length
to the three sides of the other triangle.
○Angle side angle or ASA: two angles and one side of one triangle are
equal to the corresponding two angles and side of another triangle.
○Side angle side or SAS: two sides and the included angle of one triangle
are equal to two sides and the included angle of the other triangle.
○Right angle hypotenuse Side or RHS: the hypotenuse and one side of a
right-angled triangle are equal to the hypotenuse and one other side of
the other right-angled triangle.
You can apply your knowledge of congruency to help prove other
geometrical features such as parallel and perpendicular lines.
Chapter review
1 State whether each pair of triangles is congruent or not. Give a reason for
or additional questions on
F
the topics in this chapter, visit
GCSE Mathematics Online.
your answer and give the vertices of the triangles in the correct order.
a
D
A
b
H
B
c
4 cm
C
F
3 cm
E
d
Q
A
7 cm
B
4 cm
R
8 cm
I
K
J
H
3 cm
25°
L
8 cm
30°
G
3 cm
7 cm
5 cm
30°
5 cm
4 cm
3 cm
J
G
3 cm
3 cm
L
25°
I
K
C
S
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
459
GCSE Mathematics for OCR (Foundation)
2 In the figure below, prove that angle T 5 angle R.
T
Q
P
S
R
3 Show that angle EBC 5 angle ECB in the figure below.
A
B
C
D
E
4 In the figure below, triangle ABD lies between two parallel lines.
BC 5 CA 5 AD. Prove that angle EAD 5 2 × angle ABC.
A
B
C
E
D
5 Prove that WX 5 XV in the diagram.
W
X
V
Y
460
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