properties of congruent triangles
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Properties of Congruent Triangles
Figures having the same shape and size are called congruent figures.
Are the following pairs of figures the same?
Congruence
They are the same!
If two triangles have the same shape and size, they are called congruent triangles.
Congruent Triangles and their Properties
A X
B YC Z
For the congruent triangles △ABC and △XYZ above,
A = X, B = Y, C = Z
AB = XY, BC = YZ, CA = ZX
A and X, B and Y, C and Z are called
AB and XY, BC and YZ, CA and ZX are called
A and X, B and Y, C and Z are called
A X
B YC Z
corresponding vertices.
correspondingsides.
correspondingangles.
is congruent to △XYZ△ABCThe corresponding vertices of congruent triangles should be written in the same order.
In the above example,we can also write △BAC △YXZ, but NOT △CBA △XYZ.
A X
B YC Z
The properties of congruent triangles are as follows:
AB = XY, BC = YZ,
A =X, B = Y, C = Z CA = ZX(ii) All their corresponding sides are equal.
(i) All their corresponding angles are equal,
If △ABC △XYZ …
XY
Z
4 cm
A
B
C
4.5 cm
40°
According to the properties of congruent triangles,
AB = 4.5 cm AC = 4 cm
B = 40°
△ △ A B C X Y Z
XY XZ
Y
= =
=
In the figure, △ABC △ PQR. Find the unknowns.
Follow-up question 1
A
B
C130° 30°
x cm
4 cm
7 cm
P
Q
Ry9 cm
z cm
According to the properties of congruent triangles,AC PR
CBy 180 30130180
20
4zBCQR
9x
AP
Example 1
In the figure, AB = 5 cm, AC = 4 cm and BC = 7 cm. If △ABC △DFE, find DE, EF and DF.
Solution
According to the properties of congruent triangles,
cm 5
cm 7
cm 4
ABDF
BCEF
ACDE
Example 2
In the figure, AB = 7 cm, ∠A = 50° and ∠B = 30°. If △ABC △PRQ, find PR, ∠P and ∠R.
Solution
According to the properties of congruent triangles,
30
50
cm 7
BR
AP
ABPR
Conditions for Congruent Triangles
Yes, because AB = XY, BC = YZ, CA = ZX,
∠A =∠X, ∠B =∠Y, ∠C =∠Z.
But, can we say that two triangles are congruent when only some of the properties of congruence are satisfied?
Are these two triangles congruent?
A
CB
X
Y Z
Yes… let’s see the following 5 conditions for congruent triangles first.
Condition I SSS
In △ABC and △XYZ,
if AB = XY, BC = YZ and CA = ZX,
then △ABC △XYZ.
[Abbreviation: SSS]
C
A BZ
X Y
For example,
T
U
V
2 cm
4 cm
5 cm
D
E
F 2 cm
5 cm
4 cm
TU = FE, UV = ED, TV = FD
△TUV △FED (SSS)
In △ABC and △XYZ,
if AB = XY, AC = XZ and A = X,
then △ABC △XYZ.
[Abbreviation: SAS]
Condition II SAS
C
A BZ
XY
Note that ∠A and ∠X are the included angles of the 2 given sides.
For example,
UV = FE, TV = DE, V = E
△TUV △DFE (SAS)
T
U
V
120°2 cm
2.5 cm
D
E
F2.5 cm
2 cm120°
Follow-up question 2Determine whether each of the following pairs of triangles are congruent and give the reason.
(a)A
B
C
E
G
F
3 cm
3.2 cm
3 cm
3.2 cm
2.5 cm
2.5 cm
△ABC △EGF (SSS) ◄ AB = EG, BC = GF, AC = EF
(b)I
J
K
2 cm
2.5 cm50°
N
M
L
2 cm
2.5 cm
50°
△IJK △MNL (SAS) ◄ IJ = MN, ∠J = ∠N, JK = NL
Example 3
Are △MNP and △YZX in the figure congruent? If they are, give the reason.
Yes, △MNP △YZX. (SSS)
Solution
Example 4
In the figure, WX = WY and ZX = ZY. Are △WXZ and △WYZ congruent? If they are, give the reason.
Solution
Yes, △WXZ △WYZ. (SSS)
Example 5
Which two of the following triangles are congruent? Give the reason.
Solution
△PQR △WUV (SAS)
Example 6
In the figure, AB = CD = 8 cm and ∠ABD = ∠CDB = 30°. Are △ABD and △CDB congruent? If they are, give the reason.
Yes, △ABD △CDB. (SAS)
Solution
Example 7
Which two of the following triangles are congruent? Give the reason.
Solution△DEF △ZYX (ASA)
In △ABC and △XYZ,
if A = X, B = Y and AB = XY,
then △ABC △XYZ.
[Abbreviation: ASA]
Condition III ASA
C
A B
Z
X Y
Note that AB and XY are the included sides of the 2 given angles.
U
T
V
130°20°
4 cm
D
E
F
130°20°
4 cm
For example,
U = F, UV = FD, V = D
△TUV △EFD (ASA)
Condition IV AAS
In △ABC and △XYZ,
if A = X, B = Y and AC = XZ,
then △ABC △XYZ.
[Abbreviation: AAS]
C
A B
Z
X Y
Note that AC and XZ are the non-included sides of the 2 given angles.
T
U V
D
E
F
130°
20°
20°7 cm
7 cm
For example,
U = F, TV = EDV = D,
△TUV △EFD (AAS)
130°
Follow-up question 3In each of the following, name a pair of congruent triangles and give the reason.
(a) A
BC
E
F
G
45°
40°
40°
45°
5.25 cm
5.25 cm
△ABC △FEG (ASA) ◄ ∠B = ∠E, BC = EG, ∠C = ∠G
(b)
△IJK △MNL (AAS)
K
IL
J
M
N
100°
20°
20°100°
12 cm 12 cm
B
A
C100°
12 cm
20°
◄ ∠J = ∠N, ∠K = ∠L, IK = ML
Example 8
In the figure, ∠BAD = ∠CAD and AD⊥BC. Are △ABD and △ACD congruent? If they are, give the reason.
Solution
Yes, △ABD △ACD. (ASA)
Example 9
Which two of the following triangles are congruent? Give the reason.
Solution
△PQR △ZYX (AAS)
Example 10
In the figure, ∠ABC = ∠CDA and ∠ACB = ∠CAD. Are △ABC and △CDA congruent? If they are, give the reason.
Solution
Yes, △ABC △CDA. (AAS)
In △ABC and △XYZ,
if C = Z = 90°, AB = XY and
BC = YZ (or AC = XZ),
then △ABC △XYZ.
[Abbreviation: RHS]
Condition V RHS
A
B C
X
Y Z
For example,
2 cm 2 cm 5 cm5 cm
T
U V
D
E
F
U = F = 90°, TV = ED, TU = EF
△TUV △EFD (RHS)
Are there any congruent triangles? Give the reason.
Follow-up question 4
A
B C
D
6 cm
6 cm
Yes, △ABC △ADC. (RHS) ◄ ∠B = ∠D = 90°, AC = AC, BC = DC
C
A B
Z
X Y
1. SSS
C
A B
Z
X Y
2. SAS
C
A B
Z
X Y
3. ASA
C
A B
Z
X Y
4. AAS
A
B C
X
Y Z
5. RHS
To sum up, two triangles are said to be congruent if any ONE of the following FIVE conditions is satisfied.
Example 11
Are △ABC, △RPQ and △XYZ in the figure congruent? If they are, give the reasons.
Solution
△ABC △RPQ (RHS)△XYZ △RPQ (SAS)∴ △ABC, △RPQ and △XYZ are congruent.
Example 12
In the figure, AB⊥BC, DC⊥BC and AC = DB. Are △ABC and △DCB congruent? If they are, give the reason.
Solution
Yes, △ABC △DCB. (RHS)
Properties of Similar Triangles
Similar figures have the same shape but not necessarily
the same size.
The following pairs of figures have the same shape, they are called similar figures.
Similarity
Similar Triangles and their Properties
If two triangles have the same shape, they are called similar triangles.
For the similar figures △ABC and △XYZ above,
A = X, B = Y, C = Z
AB XY
= BC YZ
CA ZX
=
AX
B YC Z
A
X
B YC Z
A and X, B and Y, C and Z are called
AB and XY, BC and YZ, CA and ZX are called
A and X, B and Y, C and Z are called corresponding vertices.
correspondingsides.
correspondingangles.
AX
B YC Z
The properties of similar triangles are as follows:
A = X, B = Y, C = Z
AB XY
= BC YZ
CA ZX
=(ii) All their corresponding sides are proportional.
(i) All their corresponding angles are equal,
~is similar to △XYZ△ABCNote: The corresponding vertices of congruent triangles should be written in the same order.
If △ABC ~ △XYZ ...
4 cm
A
B
C
4.5 cm
40°
XY
Z
2 cm
According to the properties of similar triangles,
BY 40 AC
XZABXY
cm 4cm 2
cm 5.4XY
cm 25.2 △ ~ △ A B C X Y Z
◄ XZ and AC are corresponding sides.
In the figure, △ABC ~ △PQR. Find the unknowns.
Follow-up question 5
A
B
C132° 25°
10 cm
4 cm
x cm
According to the properties of similar triangles,
P
Q
Ry5 cm
z cm4 cm
PRAC
PQAB
510
4x
8x2z
AP CBy 180
25132180 23
ACPR
BCQR
105
4z
Example 13
In △ABC and △RQP, BC = 1 cm, PQ = 2 cm, QR = 5 cm and PR = 4 cm. If △ABC ~ △RQP, find AB and AC.
Solution
According to the properties of similar triangles,
cm 5.2cm 2
cm 1
cm 5
AB
AB
QP
BC
RQ
AB
cm 2cm 2
cm 1
cm 4
AC
AC
QP
BC
RP
AC
Example 14
In the figure, AD = 3 cm, AC = 2 cm, CE = 4 cm, ∠A = 60° and BC⊥AE. If △ABC ~ △AED, find ∠E and AB.
SolutionAccording to the properties of similar triangles,
ADE = ACB = 90 In △ ADE,
∵ 180EADEA
∴
30
1809060
E
E
cm 4
cm 3
cm 2
cm )42(
AB
ABAD
AC
AE
AB
Conditions for Similar Triangles
Conditions for Similar Triangles
(i) All their corresponding angles are equal.
(ii)All their corresponding sides are proportional.
A
B CX
Y Z
We have learnt that if two triangles are similar, then
Two triangles are similar if any one of the following
three conditions is satisfied.
In △ABC and △XYZ,
if A = X, B = Y and C = Z,
then △ABC ~ △XYZ.
[Abbreviation: AAA]
Condition I AAA
A
B C
X
YZ
U = F, T = E, V = D
△TUV ~ △EFD (AAA)
For example,
T
U V
127° 25°
28°D
E
F
127°25°
28°
Condition II 3 sides prop.
In △ABC and △XYZ,
if
then △ABC ~ △XYZ.
[Abbreviation: 3 sides prop.]
,ZXCA
YZBC
XYAB
A
B C
X
Y Z
T
U
V
4 cm
2 cm
3 cm
D
E
F1.5 cm
1 cm
2 cm
For example,
△TUV ~ △DFE (3 sides prop.)
UV 2 cm FE 1 cm
= = 2, TV 3 cm DE 1.5 cm
= = 2, TU 4 cm DF 2 cm
= = 2
Condition III ratio of 2 sides, inc.
A
B C
In △ABC and △XYZ,
if and B = Y,
then △ABC ~ △XYZ.
[Abbreviation: ratio of 2 sides, inc. ]
YZBC
XYAB
X
Y Z
For example,
D
E
F
120°
2 cm
1.5 cm
V4 cm
U
T
120°3 cm
△TUV ~ △EFD (ratio of 2 sides, inc. )
UV 4 cm FD 2 cm
= = 2, UT 3 cm FE 1.5 cm
= = 2, U = F
Follow-up question 6Determine whether each of the following pairs of triangles are similar and give the reason.
(a)
E
G
F
2.4 cm
2.8 cm
2 cm
A
B
C
3 cm
3.5 cm
2.5 cm
△ABC ~ △EGF (3 sides prop.) ABEG
BCGF
◄ =ACEF
=
(b)
△ABC ~ △ZXY (AAA)
C
A B
95°
40° 45°
45°X
Y
Z40°
95°
I
J
K
2.4 cm
3 cm
50°
N
M
L
1.6 cm
2 cm
50°
△IJK ~ △MNL (ratio of 2 sides, inc. )
(c)
◄ ∠A = ∠Z, ∠B = ∠X, ∠C = ∠Y
IJMN
JKNL
◄ = , ∠J = ∠N
Example 15
Are △ABC and △XZY in the figure similar? If they are, give the reason.
Yes, △ABC ~ △XZY. (AAA)
Solution
Example 16
Are △ABC and △QRP in the figure similar? If they are,give the reason.
Solution
485082180
824850180
R
C
∴ △ ABC ~ △ QRP (AAA)
Example 17Which two of the following triangles are similar? Give thereason.
Solution
2
1
cm 10
cm 52
1
cm 8
cm 4
2
1
cm 6
cm 3
PQ
CARP
BC
QR
AB
∴ △ ABC ~ △ QRP (3 sides prop.)
Example 18In the figure, AB = 15 cm, BC = 12 cm, AC = 9 cm, BD = 20 cm and CD = 16 cm. Are △ABC and △BDC similar? If they are, give the reason.
Solution
4
3
cm 12
cm 94
3
cm 16
cm 124
3
cm 20
cm 51
CB
CADC
BCBD
AB
∴ △ ABC ~ △ BDC (3 sides prop.)
Example 19
Which two of the following triangles are similar? Give the reason.
3
2
cm 7.5
cm 5
3
2
cm 5.4
cm 3
RQ
AC
PQ
BC
∴ △ ABC ~ △ RPQ (ratio of 2 sides, inc. )
Solution
3
2
cm 7.5
cm 5
3
2
cm 5.4
cm 3
RQ
AC
PQ
BC
∴ △ ABC ~ △ RPQ (ratio of 2 sides, inc. )
Example 20
In the figure, PT = 2.7 cm, TR = 3.3 cm, QR = 3 cm, TS = 4.8 cm, RS = 2.4 cm and ∠PRQ = ∠TSR. Are △PQR and △TRS similar? If they are, give the reason.
4
5
cm 2.4
cm 34
5
cm 8.4
cm )3.37.2(
RS
QRTS
PR
∴ △ PQR ~ △ TRS (ratio of 2 sides, inc. )
Solution
Example 2 (Extra)
In the figure, △ABC △EDF and △FED △IHG. Find GH, HI and IG.
According to the properties of congruent triangles,
cm 21
cm 22
cm 20
CBFDIG
ACEFHI
BADEGH
Solution
Example 10 (Extra)
In the figure, AB = PQ, ∠ABC = ∠QRP and ∠ACB = ∠PQR. Are △ABC and △QRP congruent? If they are, give the reason.
Solution
Cannot be determined.Since the length of PR may not equal to AB.
Example 12 (Extra)In the figure, AGE, CGF and BCD are straight lines.
(a) Are △ABC and △CDE congruent? If they are, give the reason.
(b) Are △FAC and △FEC congruent? If they are, give the reason.
(a) Yes, △ ABC △ CDE. (SAS)
(b) According to the properties of congruent triangles,
AC CE
Yes, △ FAC △ FEC. (RHS)
Solution(a) Yes, △ ABC △ CDE. (SAS)
(b) According to the properties of congruent triangles,
AC CE
Yes, △ FAC △ FEC. (RHS)
(a) Yes, △ ABC △ CDE. (SAS)
(b) According to the properties of congruent triangles,
AC CE
Yes, △ FAC △ FEC. (RHS)
(a) Yes, △ ABC △ CDE. (SAS)
(b) According to the properties of congruent triangles,
AC CE
Yes, △ FAC △ FEC. (RHS)
Example 20 (Extra)
In the figure, KN = 6 cm, NM = 5 cm, LM = 4
33 cm, KM =
2
14 cm and ∠KML
= ∠KNM.
(a) Name a pair of similar triangles in the figure and give the reason.
(b) Hence, find the value of z.
In the figure, KN = 6 cm, NM = 5 cm, LM = 4
33 cm, KM =
2
14 cm and ∠KML
= ∠KNM.
(a) Name a pair of similar triangles in the figure and give the reason.
(b) Hence, find the value of z.
In the figure, KN = 6 cm, NM = 5 cm, LM = 4
33 cm, KM =
2
14 cm and ∠KML
= ∠KNM.
(a) Name a pair of similar triangles in the figure and give the reason.
(b) Hence, find the value of z.
In the figure, KN = 6 cm, NM = 5 cm, LM = 4
33 cm, KM =
2
14 cm and ∠KML
= ∠KNM.
(a) Name a pair of similar triangles in the figure and give the reason.
(b) Hence, find the value of z.
In the figure, KN = 6 cm, NM = 5 cm, LM = 4
33 cm, KM =
2
14 cm and ∠KML
= ∠KNM.
(a) Name a pair of similar triangles in the figure and give the reason.
(b) Hence, find the value of z.
Solution
(a)
4
3
cm 6
cm 2
14
4
3
cm 5
cm 4
33
NK
MK
MN
LM
∴ △ KLM ~ △ KMN (ratio of 2 sides, inc. )
(a)
4
3
cm 6
cm 2
14
4
3
cm 5
cm 4
33
NK
MK
MN
LM
∴ △ KLM ~ △ KMN (ratio of 2 sides, inc. )
(b) According to the properties of similar triangles,
8
33
cm 6
cm 2
14
cm 2
14
cm
z
z
NK
MK
KM
KL
(b) According to the properties of similar triangles,
8
33
cm 6
cm 2
14
cm 2
14
cm
z
z
NK
MK
KM
KL
(b) According to the properties of similar triangles,
8
33
cm 6
cm 2
14
cm 2
14
cm
z
z
NK
MK
KM
KL
(b) According to the properties of similar triangles,
8
33
cm 6
cm 2
14
cm 2
14
cm
z
z
NK
MK
KM
KL
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