1. What are you going to
learn? Properties of Two Congruent Figures
to identify two congruent
or incongruent plane
figures by showing the
requirements
11..33
Examine two one-thousand-rupiah notes
(paper money). You will find that they are the
same in both shape and size. In other words,
they are said to be congruent.
to determine congruent
triangles
to prove two congruent
triangles Now look at the quadrilaterals below.
to determine the
proportions of the sides of
two congruent triangles
and to find their lengths
A
B
C
P
Q
S
D
R
Figure 1.7
to show the
consequences of two
congruent triangles
to distinguish the concept
of similarity from that of
congruency
Key term:
• congruent
Discussion
a. What are the side lengths of quadrilaterals ABCD and PQRS? Measure the
corresponding sides with a ruler.
b. What are the sizes of the angles of quadrilaterals ABCD and PQRS?
Measure the corresponding angles with a protractor.
c. Are the two figures congruent? Explain.
d. What are the requirements for two polygons to be congruent? Explain.
e. Find objects around you that have congruent surfaces.
Using the requirements for two polygons to be congruent, find the pairs of
congruent figures below.
Mathematics for Junior High School Grade 9 / 17

2. Real Life Situation
Now look at the patterns of ties below.
Figure 1.9 shows examples of the geometrical patterns.
(a) (b)
Figure 1.8
Figure 1.9
D
A B
C
E
F
G
H
The ties above have triangular patterns. The shape and the size of the
triangles on each tie are the same. Such triangles are examples of congruent
triangles. For further clarification, examine the following explanation.
Activities
Draw two rectangles as the figures on
the left. Move rectangle ABCD to the
right so that point A coincides with
point K, B with L, C with M, and D with
N. Rectangle ABCD, then, will precisely
cover rectangle KLMN. In such case,
ABCD is said to be congruent to KLMN.
It is symbolised by ABCD ≅ KLMN.
A B
CD
K L
MN
Figure 1.10
18 / Student’s Book – Similarity and Congruency

3. P
R
Q F
G
Figure 1.11
S
Copy trapezium PQRS (Figure 1.11)
on a piece of paper and cut it. Move it
so that P coincides with E, Q with F,
R with G, and S with H. Then
trapezium PQRS covers trapezium
EFGH. In other words, PQRS is
congruent to EFGH. This is
symbolised by PQRS ≅ EFGH.
E
H
Copy ΔABC (Figure 1.12) on a piece
of paper and cut it. Move it so that A
coincides with P, B with Q, and C
with R. In other words, ΔABC covers
ΔPQR or ΔABC is congruent to ΔPQR.
This is symbolised by ΔABC ≅ ΔPQR.
The following are examples of congruent triangles.
Look at Figure 1.13. Which triangles are congruent to ΔABC? Explain what
you are going to do to ΔABC to make it cover the triangles congruent to it.
Solution:
The triangles congruent to ΔABC are ΔJIH
and ΔMKL.
ΔABC will precisely cover ΔJIH if you
make the right movement.
ΔABC will also precisely cover ΔMKL if
you make the right movement.
A
C
Q P
R
B
Figure 1.12
Mathematics for Junior High School Grade 9 / 19

4. Real life example
Look at the figure of a tent below.
The front part of the tent has a triangular
shape.
A
PC M
Is ΔACP ≅ ΔAMP? Explain.
Solution:
ΔACP ≅ ΔAMP, because ΔACP can
precisely cover ΔAMP by reflecting
ΔACP on AP .
20 / Student’s Book – Similarity and Congruency

5. Characteristics of Two Congruent Triangles
Look at the figure of a bridge on the
left. To make it stronger, the bridge is
supported by iron bars forming
triangles. Look at ΔMPO and ΔNQK. If
the picture is redrawn and enlarged,
ΔMPO and ΔNQK will look like the
triangles in Figure 1.15.
Figure 1.14
P
N
M
O Q K
Figure 1.15
Move ΔMPO so that it precisely covers ΔNQK. Then the two triangles
are congruent. PO coincides with QK, PM with QN, and OM with KN. The
sides that coincide are called the corresponding sides. So, sides PO and
QK are corresponding sides, PM and QN are corresponding sides, and OM
and KN are corresponding sides.
Two congruent triangles have the
corresponding sides of the same length
Two congruent
triangles
Mathematics for Junior High School Grade 9 / 21

6. Congruency of figures can be determined from the size of corresponding
angles. Look at the following congruent triangles.
C
B
A
R
P
Q
Since ΔABC ≅ ΔPQR, both triangles can cover one to each other.
Consequently, point A coincides with P, point B with Q, and point C with R,
and ∠CAB = ∠RPQ, ∠ABC = ∠PQR, and ∠ACB = ∠PRQ. Therefore, ∠CAB and
∠RPQ are corresponding angles, ∠ABC and ∠PQR are corresponding angles,
and ∠ACB and ∠PRQ are corresponding angles.
Two congruent triangles have the
corresponding angles of the same size
Two congruent
triangles
EXAMPLE 2
ΔUVW and ΔDEF below are congruent. Find the sides of the
same length and the angles of the same size.
Solution:
Since ΔUVW and ΔDEF are congruent, the
corresponding sides are the same length, or
UV = DE, UW = DF dan VW = EF.
Furthermore, the corresponding angles are the
same size, or
∠U = ∠D, ∠V= ∠E, and ∠W = ∠F.
D
E
F
U
V
W
Figure 1.16
22 / Student’s Book – Similarity and Congruency

7. Requirements and Consequences of Congruency
Look at the following figures.
AB = PQ, AC = PR dan BC = QR.
Move ΔABC so that point A will
coincide with P, point B with Q,
and point C with R, so that ΔABC
will precisely cover ΔPQR.
Therefore, ΔABC ≅ ΔPQR.
Two triangles will be congruent if
the three corresponding sides are
the same length.
What are the consequences if two triangles are congruent according to
(s, s, s)? Look at the two figures below.
Two triangles are congruent if the three sides of
the first triangle are the same length as the
corresponding sides of the second triangle (s, s, s)
Requirements for
two congruent
triangles
A
B
P
Q
C R
Figure 1.17
RC
**
A
B
P
Q
Figure 1.18
Mathematics for Junior High School Grade 9 / 23

8. The figures show that ΔABC and ΔPQR have two sides of the same
length, and the angles between the sides are the same size. Then
AB = PQ, AC = PR, and ∠A = ∠P
Suppose we move ΔABC such that point A coincide with P.
Because ∠A = ∠P, then ∠A coincides with ∠P. Because AC = PR, point
C coincides with R, and because AB = PQ, point B coincides with Q.
ΔABC will then precisely cover ΔPQR. Therefore, ΔABC and ΔPQR are
congruent. Two triangles will be congruent if the two sides are the same
length and the angles between the sides are the same size.
Two triangles will be congruent if the two sides
of the first triangle are the same size as the
corresponding sides of the second triangle, and the
angles between the sides are the same size (s, a, s).
Requirements for
two congruent
triangles
What are the consequences of the two triangles being congruent
according to (s, a, s)?
We have identified two requirements for two congruent triangles. For
the third requirement, we are going to look at two triangles that have one
corresponding side of the same length and the two corresponding angles
on the corresponding sides are the same size.
Look at Figure 1.19 below.
A
B
C
x
P
Q
R
x
y
y
Figure 1.19
24 / Student’s Book – Similarity and Congruency

9. The figure shows that
∠A = ∠P, AB = PQ, and ∠B = ∠Q.
AB is the side where ∠A and ∠B are located.
PQ is the side where ∠P and ∠Q are located.
Since the sum of the three angles in a triangle is 180°,
∠A + ∠B + ∠C = 180° and
∠P + ∠Q + ∠R = 180°
Therefore,
∠C = 180° - ∠A – ∠B, and
∠R = 180° - ∠P – ∠Q.
Because ∠A = ∠P and ∠B = ∠Q, ∠R = 180° - ∠A - ∠B
So, ∠C = ∠R
Finally, we have the following relationship
∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.
Therefore, the three corresponding angles in the two triangles are the
same size.
By definition, the two triangles are similar. Since the two triangles are
similar, the proportions of the corresponding sides are the same, namely
PQ
AB =
QR
BC =
PR
AC
Since AB = PQ (given), then
PQ
AB =
QR
BC =
PR
AC = 1
Accordingly, AB = PQ, BC = QR and AC = PR. This means that the three
corresponding sides in the two triangles are the same length. By
requirement (s, s, s) that we have learned, then ΔABC ≅ ΔPQR. What are
the consequences?
Mathematics for Junior High School Grade 9 / 25

10. Two triangles will be congruent if two angles in
the first triangle are the same size as the
corresponding angles in the second triangle,
and the common leg of the two angles are the
same length (a, s, a).
Requirements
for two
congruent
triangles
Investigate whether ΔRQT and ΔSQT in Figure
1.20 are congruent. Look closely at ΔRQT and
ΔSQT. What are the consequences?
Answer
The figure shows that RT = ST, RQ = SQ, and
TQ = TQ, or the three corresponding sides in
the two triangles are the same length. By the
requirement (s, s, s), ΔRQT ≅ ΔSQT.
Consequently, ∠R = ∠ S, ∠RTQ = ∠STQ, and
∠TQR = ∠TQS.
∴
Investigate whether ΔDAC and ΔBAC are
congruent. What are the consequences?
Answer
Look closely at ΔDAC and ΔBAC. It shows
DA = BA, ∠DAC = ∠BAC, and AC = AC. By
the requirement (s,a,s), ΔDAC ≅ ΔBAC.
Consequently, CD = BC, ∠ADC = ∠ABC, and
∠DCA = ∠BCA
T
6 m 6 m
Q
R S2 m 2 m
Figure 1.20
A
Figure 1.21
BD
C
3 cm3 cm
O O
26 / Student’s Book – Similarity and Congruency

11. Figure 1.22 shows that ∠A = ∠M and ∠B =
∠L, such that ΔABC ≅ ΔMLK.
K
Reason:
Because ∠A = ∠M, AB = ML, and ∠B = ∠L,
by the requirement (a, s, a), ΔABC ≅ ΔMLK.
As a result,
∠B = ∠K, BC = KL, and AC = KM.
B
C
A
L
M
4 cm
4 cm
Figure 1.22
Given parallelogram ERIT on the right.
Show that TP = RO.
Solution:
E
R
T
I
P
O
To show that TP = RO, complete the blank
spaces on the left column.
Statements Reasons
Look at ΔTIE and ΔREI
1. IT = ER, ET = IR, EI = IE 1. Given
2 a. ΔTIE ≅ Δ . . .
b. ∠TIE = ∠ … and
∠TEI = ∠ …
2 a. (s,s,s)
b. corresponding angles
Now look at ΔTPE and ΔROI.
3. ∠ TPE =∠ … 3. both having a size of 900
4. ∠TEP = ∠ … 4. according to 2b
5. ∠PTE = 900 - ∠TEP 5. the sum of three angles in a triangle
being 1800
6. ∠ORI = 900 - ∠ … 6. the sum of three angles in a triangle
being 1800
7. ∠PTE = ∠ORI 7. according to 5 dan 6
∠TEP = ∠RIO, ET = RI, and ∠PTE = ∠ORI. By the requirement (…, … , …),
ΔTDE ≅ ΔROI. Because TP and RO are corresponding sides, then TP = RO.
Mathematics for Junior High School Grade 9 / 27

12. Similarity and Congruency
Look at the following two equilateral triangles.
C
A B
R
P Q
Figure 1.23
a. Are ABC andΔ
Δ PQR similar?
Explain.
b. Are ABC andΔ
Δ PQR congruent?
Explain.
c. Are two similar
triangles always
congruent? Explain.
Look at the following two triangles
A
C
Q
R
B
P
a. Are ΔABC and Δ PQR
similar? Explain.
b. Are ΔABC and Δ PQR
congruent? Explain.
c. Are two congruent
triangles always similar?
Explain.
Figure 1.24
If two triangles are congruent, then they are similar.
If two triangles are similar, they are not necessarily congruent
28 / Student’s Book – Similarity and Congruency

13. Find the pairs of congruent triangles and the pairs of similar triangles in Figure
1.25 below.
1 2 3
4
5
6
7
8
9
10
Figure 1.25
1. Measure the following figures then determine whether the triangles in each
pair are congruent. If they are, give your reasons and find the
corresponding sides and the corresponding angles.
a. b.
C
A
B
O
M
R
K
W
V
U
T
L
2. Are the triangles in each pair below congruent? If they are, give your
reasons. What are the consequences?
a. A G b. 3
T 7
7
N 5 3 5
R
M
Mathematics for Junior High School Grade 9 / 29

14. c. A d.
G
E
C
G E
A
C B
D
3. Explain why ΔBDF ≅ ΔMKH, then find the values of m, s, and n.
K
M
s
t
8
72O
nO
32O
H
D
B
9
t
872O
nO
mO
F
4. Are ΔFKL and ΔKFG congruent? Give your reasons. If they are, find the
corresponding sides and the corresponding angles.
F G
L
K
5. PQRS is a kite. Show ΔPQR and ΔPSR are congruent triangles. Find the
sides of the same length and the angles of the same size.
P
S
R
Q
30 / Student’s Book – Similarity and Congruency

15. For questions 6-13, use requirements (s,s,s), (s,a,s) or (a,s,a) to verify each
statement.
6. AB = CB 7. ∠OME = ∠ERO
B
M O
A C
E R
D
8. ∠TSP = ∠TOP 9. KP =LM
S O K L
T
P P M
Y
10. ∠ORE = ∠OPE 11. CT = RP
O
R P C R
E
N
T P
Mathematics for Junior High School Grade 9 / 31

16. 12. If line l is perpendicular to AB and CA = CB, show that PA=PB.
l
A B
P
C
13. Suppose ABCD is a parallelogram. Show that ΔABC ≅ ΔCDA.
D C
A
B
32 / Student’s Book – Similarity and Congruency