1. science
2. mathematics
3. geometry

NSM Enhanced 10 5.1–5.2
Chapter Focus
Congruent
Triangles
3
In this chapter students are encouraged to
use deductive geometry to construct proofs
of geometrical relationships involving
congruent triangles, and to prove properties
of special triangles and quadrilaterals.
Learning Outcomes
SGS5.2.2 Develops and applies results for
proving that triangles are
congruent or similar.
Chapter Contents
ge
s
Be careful dear–
it could be a scam!
pa
3:01 Congruent triangles
SGS5·2·2
3:02 Applying the congruency tests
SGS5·2·2
Fun Spot: What do you call a man with
a shovel?
3:03 Using congruent triangles to find
sides and angles
SGS5·2·2
3:04 Deducing properties of the special
triangles
SGS5·2·2
3:05 Deducing properties of the special
quadrilaterals
SGS5·2·2
Maths Terms, Diagnostic Test, Revision
Assignment, Working Mathematically
Learning Outcomes
SGS5·2·2 Develops and applies results for proving that triangles are congruent or similar.
Sa
m
pl
e
Working Mathematically Stages 5·2·1–5
1 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting
62
Vocabulary Preview
congruent triangles
deducing
definition
matching angles
proof
quadrilateral
triangle
62
New Signpost Mathematics Enhanced 10 5.1–5.2 ``TEACHER EDITION
Chapter 3
3:01 Congruent Triangles
Learning Outcomes
Outcome SGS5·2·2
SGS5.2.2 Develops and applies results for
• Congruent figures are the same shape and size. When one is superimposed on the other, they
coincide exactly.
• Congruent figures are produced by translations, reflections and rotations.
• When congruent figures are placed on top of each other so that they coincide exactly, the
matching sides and angles are obviously equal. The word corresponding is often used instead
of matching.
proving that triangles are
congruent or similar.
Knowledge and Skills
Students learn about:
Prep Quiz 3:01
A
•
B
C
determining what information is needed
to show that two triangles are
congruent:
−
If three sides of one triangle are
respectively equal to three sides of
another triangle, then the two
triangles are congruent (SSS).
−
If two sides and the included angle
of one triangle are respectively
equal to two sides and the included
angle of another triangle, then the
two triangles are congruent (SAS).
D
1 Which figure is congruent to figure A?
2 Which figure is congruent to figure B?
C
O
M
N
The figures to the left are congruent.
5 Name the angle that matches ∠B.
6 Name the side that matches AB.
7 Name the angle that matches ∠N.
O
E
D
P
B
N
L
C
M
Are the following pairs of triangles congruent?
9
8
ge
B
A
P
D
pa
E
The figures to the left are congruent.
3 Name the angle that matches ∠A.
4 Name the side that matches FE.
Q
s
L
F
A
Congruent triangles
Sa
m
pl
• In geometry, we are often asked to show that two sides or two angles are equal. A common way
of doing this is by showing that they are the matching sides or angles of congruent triangles.
• To check that two triangles are congruent, we would normally need to compare six pieces of
information (three sides and three angles).
• In the next exercise we will investigate the minimum conditions for congruent triangles.
A minimum condition is the smallest amount of information that we need to know about
two triangles before we can say they are congruent.
Answers
Chapter 3 Congruent Triangles
Technology
Prep Quiz 3:01
1
4
7
10
figure C
QP
∠C
no
2 figure D
5 ∠O
8 no
If two angles and one side of one
triangle are respectively equal to
two angles and the matching side of
another triangle, then the two
triangles are congruent (AAS).
−
If the hypotenuse and a second side
of one right-angled triangle are
respectively equal to the hypotenuse
and a second side of another rightangled triangle, then the two
triangles are congruent (RHS).
e
10
−
3
6
9
∠L
PO
no
•
63
applying congruent triangle results to
establish properties of isosceles and
equilateral triangles eg
−
If two sides of a triangle are equal,
then the angles opposite the equal
sides are equal. Conversely, if two
angles of a triangle are equal, then
the sides opposite those angles are
equal.
−
If three sides of a triangle are equal
then each interior angle is 60º.
Chapter Link
Exercise 1:04 Geometry can be used as a
Pre-Topic Test for this chapter.
Class Tutorial:
Congruency
Interactive:
Congruent triangles
Chapter 3 Congruent Triangles
63
NSM Enhanced 10 5.1–5.2
Exercise 3:01
5 cm
3
m
3c
cm
60°
50°
4 cm
50°
60°
What is the
least I need
to know?
60°
ge
3 cm
s
3 cm
Use the diagrams above to answer questions 1 and 2.
2
Are two triangles congruent if they:
a have only one side equal?
c have only two sides equal?
e have one side and one angle equal?
b have only one angle equal?
d have two angles equal?
pa
1
Can we be sure that two triangles are congruent if we can compare only two pieces of
information on each one?
m
pl
e
To compare three pieces of information we could compare:
• three sides
• two sides and one angle
• one side and two angles
• three angles
Sa
3
64
a When a photograph is enlarged, are:
i the angles in the photo and enlargement the same?
ii the photo and the enlargement congruent?
b If two triangles have their three angles equal, does it mean they are congruent?
New Signpost Mathematics Enhanced 10 5.1–5.2
Answers
Exercise 3:01
Teacher’s notes
1 No in all cases.
2 We cannot say that the triangles are
definitely congruent.
3 a
b
4 a
b
c
5 a
b
6 a
64
i yes
ii no
no
yes
They are congruent.
yes
They are congruent.
∠A and ∠E, ∠B and ∠D, ∠C and ∠F
yes
b no
New Signpost Mathematics Enhanced 10 5.1–5.2 ``TEACHER EDITION
Chapter 3
4
Teaching Strategies
and Ideas
Copy one of the following triangles onto paper and cut it out.
C
F
Exercise 3:01 can be used to develop the
four tests for congruent triangles.
A
D
B
E
•
Questions 4 and 5 show that two
triangles are congruent if three sides of
one triangle are equal to three sides of
the other (SSS).
•
Questions 6 to 9 show that two
triangles are congruent if two sides and
the included angle of one triangle are
equal to two sides and the included
angle of the other (SAS).
•
Questions 10 and 11 show that two
triangles are congruent if two angles
and a side of one triangle are equal to
two angles and the matching side of the
other (AAS)
a Do the triangles have their sides equal in length?
b By superimposing the cut-out triangle over the other triangle, see if the triangles are
congruent.
c Do the triangles have the same sized angles?
5
Construct or trace one of the triangles below and cut it out.
F
A
5 cm
B
6 cm
4 cm
6 cm
C
D
4 cm
E
5 cm
s
a By placing the cut-out triangle over the other triangle, find out if the two triangles are
congruent.
b Name the pairs of matching (or corresponding) angles.
ge
•
pa
In questions 4 and 5, the triangles with matching sides of equal length were congruent.
We say that three pairs of sides equal is a minimum condition for congruent triangles. It is
abbreviated to SSS.
6
60°
3 cm
Questions 12 and 13 show that two
right-angled triangles are congruent if
the hypotenuse and one other side are
equal to the hypotenuse and one side of
the other triangle (RHS).
3 cm
4 cm
4 cm
e
60°
Sa
m
pl
a Do the triangles above have two sides and one angle equal?
b Are the triangles congruent? (Check by measuring the third side.)
Chapter 3 Congruent Triangles
65
Teacher’s notes
Chapter 3 Congruent Triangles
65
NSM Enhanced 10 5.1–5.2
The diagram shows a triangle with sides AB and AC of given lengths. The angle CAB is allowed
to vary in size so that C moves on a circle, radius AC, centre A.
a What happens to the length of BC as ∠BAC
C
gets bigger?
C
b If you fix ∠BAC at a certain size (say 30°) is it
possible to get two different lengths for BC?
C
cm
3 cm
3c
I see! The angle size
determines the length
of BC.
m
A
Here is a game for students: Student A
constructs a triangle labelling each side
and angle. Student A tells student B
three pieces of information about their
triangle and student B must construct
an identical triangle. Students can take
turns explaining and reconstructing.
Construct or trace the triangle shown on a piece
of paper.
a Measure BC to the nearest millimetre.
b Is it possible to get more than one triangle
from this information?
c Cut your triangle out and compare it with
the triangles of other students in your class.
Are they all congruent?
Yes, and for each angle
size, there is only one
corresponding length.
C
4 cm
A
60°
5 cm
B
s
8
B
2 cm
Construct or trace one of the triangles below and cut it out.
ge
9
A
3 cm
pa
E
5 cm
D
120°
3 cm
120°
B
5 cm
C
F
e
a Is AC = EF?
b By superimposing, find if ΔABC is congruent to ΔDEF.
In questions 6 to 9, we found that when two sides and the angle between them in
one triangle are equal to two sides and the angle between them in the other triangle then the
triangles are congruent.
m
pl
•
The class can discuss the difference
between SAS and RHS, since both
involve two sides and an angle being
equal. The question could be posed:
Isn’t RHS just a special case of SAS?
This can be used to reinforce that the
angle must be between the sides for SAS.
It is important that the angle is included by (ie between) the two sides. This is a minimum
condition for congruent triangles, and is abbreviated to SAS.
Sa
•
7
3
Teaching Strategies
and Ideas
66
New Signpost Mathematics Enhanced 10 5.1–5.2
Answers
Exercise 3:01
7 a BC gets longer.
b no
8 a 4·6 cm b no
c yes
9 a yes
b The triangles are congruent.
10 a I and V, II and IV, III and VI
b The 2.5 cm side is opposite the same
sized angle.
11 a
i
80°
ii 60°
Teacher’s notes
iii 40°
b ΔDEF
c The 4 cm side is opposite the 60° angle
in both triangles.
66
New Signpost Mathematics Enhanced 10 5.1–5.2 ``TEACHER EDITION
Chapter 3
10
Real World Application
In each of the following triangles, the angles match, and one side in each has the same length.
Construct or copy each triangle and cut it out.
50°
60°
70°
I
II
50°
60°
III
70°
2·5 cm
Synchronised swimming is where two
swimmers complete the same moves at the
same time. The aim is that they appear to be
exactly the same; that is, they are congruent
with each other.
50°
2·5 cm
70°
60°
2·5 cm
The sides opposite
equal angles are
matching sides.
50°
60°
70°
VI
IV
V
50°
70°
50°
2·5 cm
60°
70°
60°
2·5 cm
2·5 cm
a Which triangles are congruent?
b For each pair of congruent triangles, how could you describe the position of the 2.5 cm side?
Construct or copy each triangle and cut it out.
L
Identical twins could be called congruent
twins if they are the same shape and size.
s
40°
D
A
60°
40°
4 cm
C
80°
E
40°
4 cm
F
80°
M
a Which angle is the 4 cm side opposite in:
i ΔABC?
ii ΔDEF?
iii ΔLMN?
b Which of the triangles shown is congruent to ΔXYZ?
c How could you describe the position of the
4 cm side in each of the congruent triangles?
pa
B
ge
60°
80°
60°
N
4 cm
m
pl
e
X
Y
40°
4 cm
60°
80°
Z
Questions 10 and 11 have shown us that if the angles of one triangle are equal to the
angles of another triangle, and a side in one is equal to a side in the same position of the
other, then the triangles are congruent. This is the third minimum condition, and is
abbreviated to AAS.
Sa
11
Teacher’s notes
Chapter 3 Congruent Triangles
67
Homework 3:01
Chapter 3 Congruent Triangles
67
NSM Enhanced 10 5.1–5.2
Facts
12
The symbol G was first used to represent
congruence by Gottfried Leibniz in the
1679. By the 1800s, the symbol had
evolved to H and today we use o.
The fourth set of minimum conditions is restricted
to right-angled triangles only.
Copy one of the triangles below and cut it out.
Does your cut-out match both triangles?
C
The hypotenuse is the longest
side. For this reason,
Pythagoras’ theorem says:
c2 = a2 + b2
D
5 cm
4 cm
c
b
E
4 cm
A
5 cm
B
a Write down the pairs of matching sides.
b Are the triangles congruent?
13
F
a
The two right-angled triangles shown have their hypotenuse and one side equal in length.
a Write down Pythagoras’ theorem for
each triangle.
m
c
b
a
b By rearranging the formula, show
that a = m.
c
c Are the triangles congruent?
b
ge
s
Questions 12 and 13 have shown us that two right-angled triangles are congruent if the
hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the
other triangle. This is the fourth condition, and is abbreviated to RHS.
Sa
m
pl
e
pa
Summary
• Two triangles are congruent if three sides of one
triangle are equal to three sides of the other. (SSS)
• Two triangles are congruent if two sides and the
included angle of one triangle are equal to two sides
and the included angle of the other. (SAS)
• Two triangles are congruent if two angles and a side
of one triangle are equal to two angles and the
matching side of the other. (AAS)
• Two right-angled triangles are congruent if the
hypotenuse and one side of one triangle are equal
to the hypotenuse and one side of the other
triangle. (RHS)
• The symbol ≡ means ‘is congruent to’.
Answers
68
New Signpost Mathematics Enhanced 10 5.1–5.2
Teacher’s notes
Exercise 3:01
12 a
b
13 a
b
BC and EF, AB and DF, AC and DE
yes
c2 = a2 + b2; c2 = m2 + b2
a2 = c2 − b2 and m2 = c2 − b2, ∴ a2 = m2,
∴ a = m (as both are positive)
c yes
68
New Signpost Mathematics Enhanced 10 5.1–5.2 ``TEACHER EDITION
• SSS means
‘side, side, side’.
• SAS means
‘side, angle, side’.
• AAS means
‘angle, angle, side’.
• RHS means
‘right angle,
hypotenuse, side’.
Chapter 3
Learning Outcomes
3:02 Applying the
Congruency Tests
Outcome SGS5·2·2
SGS5.2.2 Develops and applies results for
proving that triangles are
congruent or similar.
Prep Quiz 3:02
D
A
5
B
5
3
6
C
6
Name the side that corresponds to:
1 AC
2 AB
3 BC
Name the angle that corresponds to:
4 ∠A
5 ∠B
6 ∠C
Knowledge and Skills
Students learn about:
E
3
F
P
L
Name the side that corresponds to:
7 LM
8 MN
M
N
Q
A
x°
60°
70° 50°
Z
2 cm
B
A
8 cm
5 cm
C
5 cm
F
Solutions
■ Note: ≡ means
2 Which of the congruency tests can be
used to show that ΔABC is congruent
to ΔDBC?
‘is congruent to’.
s
If ABC is
congruent
to DEF,
we write
ABC DEF
e
D
m
pl
2
C
A
B
3 Are these two triangles congruent?
Z
A
B
70°
60°
A
4 cm
50°
C
X
60° 70°
Y
B
Chapter 3 Congruent Triangles
applying congruent triangle results to
establish some of the properties of
special quadrilaterals, including
diagonal properties eg the diagonals
of a parallelogram bisect each other
Working Mathematically
Students learn to:
•
apply the properties of congruent and
similar triangles to solve problems,
justifying the results (Applying
Strategies, Reasoning)
•
apply simple deductive reasoning in
solving numerical and non-numerical
problems (Applying Strategies, Reasoning)
D
The red markings show the equal sides and
equal angle. ΔABC and ΔDCB have two
sides and an included angle equal.
∴ ΔABC ≡ ΔDCB (SAS)
continued ➜➜➜
Sa
4 cm
•
1 ΔABC and ΔDEF have
all their sides equal.
∴ ΔABC ≡ ΔDEF (SSS)
6 cm D
C
ge
E
pa
1 Why is ΔABC congruent to ΔDEF?
8 cm
applying congruent triangle results to
establish properties of isosceles and
equilateral triangles eg
− If two sides of a triangle are equal,
then the angles opposite the equal
sides are equal. Conversely, if two
angles of a triangle are equal, then
the sides opposite those angles are
equal.
− If three sides of a triangle are equal
then each interior angle is 60º.
50° 60°
2 cm C
Worked examples
B
•
9 Find the value of x.
10 Are the 2-cm sides corresponding?
The minimum conditions deduced in the last section are used to prove that two triangles are
congruent. Special care must be taken in exercises that involve overlapping triangles.
6 cm
applying the congruency tests to justify
that two triangles are congruent
R
X
Y
•
69
Teacher’s notes
Answers
Prep Quiz 3:02
1
4
7
10
EF
∠E
PQ
no
2 ED
5 ∠D
8 QR
3 DF
6 ∠F
9 70
Chapter 3 Congruent Triangles
69