Download Chapter 10 Congruent and Similar Triangles

Chapter 10
Congruent and Similar
Triangles
Introduction
Recognizing and using congruent and
similar shapes can make calculations and
design work easier. For instance, in the
design at the corner, only two different
shapes were actually drawn. The design
was put together by copying and
manipulating these shapes to produce
versions of them of different sizes and in
different positions.
In this chapter, we will look in a little more
depth at the mathematical meaning of the terms
similar and congruent, which describe the relation
between shapes like those in design.
Similar and Congruent Figures
• Congruent polygons have all sides
congruent and all angles congruent.
• Similar polygons have the same shape;
they may or may not have the same size.
Worksheet :
Exercise 1 : Which of the following pairs are congruent
and which are similar?
Examples
These figures are
similar and congruent.
They’re the same shape
and size.
These figures are similar
but not congruent.
They’re the same shape,
but not the same size.
Another Example
These figures are neither
similar nor congruent.
They’re not the same
shape or the same size.
Even though they’re both
triangles, they’re not
similar because they’re
not the same shape
triangle.
Note: Two figures can be similar but not
congruent, but they can’t be congruent but not
similar. Think about why!
Congruent Figures
When 2 figures are congruent, i.e.
2 figures have the same shape
and size,
Corresponding angles are equal
Corresponding sides are equal
Symbol : 
Congruent Triangles
A
ABC  XYZ
X
B
Z
C
Y
• AB = XY, BC = YZ, CA = ZX
• A =X , B =Y, C =Z
Note : Corresponding vertices are named in order.
THE ANGLE MEASURES OF A TRIANGLE AND
CONGRUENT TRIANGLES
•
The sum of the angle measures of a triangle is 180o
Example
?
65o
?=
•
30o
85o
Congruent triangles
Congruent triangles are triangles with the same shape and size
Angle = 60o; side = 5cm
5 cm
Example
60o
?
90o
?
Isosceles triangles
• An isosceles triangle is the triangle which
has at least two sides with the same length
• In an isosceles triangle, angles that
are opposite the equal-length sides have
the same measure
?
Example
82cm
?
52o
The side = 82 cm, the angle = 76o
Equilateral triangles
• An equilateral triangle has three sides of equal
length
• In an equilateral triangle, the measure of
each angle is 60o
Example
60o
?
100cm
?
Angle = 60o, side = 100 cm
Right triangles and Pythagorean theorem
• A right triangle is the triangle with one right angle
Hypotenuse
c
Leg
• Pythagorean theorem
b
c2 = a2 + b2
Leg
a
Example
60o
?
3 cm
?
4 cm
c2 = 42 + 32 = 25
C=5
Ex 10A Page 47
• Q2 a
• By comparing,
x = 4.8,
y = 42
• Q2 d
• By comparing,
x = 22,
y = 39 – 22
= 17
• Q2 b
• By comparing,
x = 16,
y = 30
( 180- 75- 75)
Tests for Congruency
Ways to prove triangles congruent :
• SSS ( Side – Side – Side )
• SAS ( Side – Angle – Side )
• ASA ( Angle – Side – Angle ) or AAS (
Angle –Angle – Side )
• RHS ( Right angle – Hypotenuse – Side )
SSS ( Side – Side –Side )
• Three sides on one triangle are equal to
three sides on the other triangle.
A
B
C
X
•AB = XY,
•BC = YZ,
•CA = ZX
ABC  XYZ
(SSS)
Y
Z
Example :
A
C
B
D
Given AB = DB and AC = DC.
Prove that ABC  DBC
• AB = DB ( Given )
• AC = DC ( Given )
• BC ( common)
• Hence ABC  DBC ( SSS )
Textbook Page 44 Ex 10A Q 1 a, k
SAS ( Side – Angle – Side )
• Two pairs of sides and the included angles
are equal.
A
B
C
X
Y
•AB = XY,
•BC = YZ,
•ABC = XYZ
( included angle )
ABC  XYZ
(SAS)
Z
Example : B
E
A
C
Given AC = EC and BC = DC.
D
Prove that ABC  EDC
• AC = EC ( Given )
• ACB = ECD ( included angle, vert opp )
• BC = DC ( Given )
• Hence ABC  EDC ( SAS )
Textbook Page 44 Ex 10A Q 1 c, i
ASA ( Angle – Side – Angle )
AAS ( Angle – Angle – Side )
• Two pairs of angles are equal and a pair of
corresponding sides are equal.
A
B
X
Y
C
•AB = XY,
•ABC = XYZ
•BAC = YXZ
ABC  XYZ
(ASA)
From given diagram, ACB = XZY
Z
ABC  XYZ (AAS)
Example : B
E
A
C
Given AC = EC and BAC = DEC
Prove that ABC  DEC
• AC = EC ( Given )
• BAC = DEC ( Given )
• ACB = ECD (vert opp)
• Hence ABC  EDC ( ASA )
Textbook Page 44 Ex 10A Q 1 f, o
D
RHS
( Right angle – Hypotenuse – Side )
A
• Right-angled triangle with the hypotenuse
equal and one other pair of sides equal.
C
B
Z
•ABC = XYZ = 90°
( right angle)
•AC = XZ ( Hypotenuse)
•BC = YZ
ABC  XYZ (RHS)
Y
X
Example :
A
B
C
Prove that ABC  DBC
• ACB = DCB = 90
• AB = DB ( Given, hypotenuse )
• BC is common
Hence ABC  EBC ( RHS )
Textbook Page 44 Ex 10A Q 1 g, j
Try Q1 e , 1y too
D
Time to work
•
•
•
•
•
•
Class Work
Ex 10B Pg 49
Q1
Q2
Q4
Q6
• Home Work
• Ex 10A Page 4447
• Q 1 b, h, m, p, r, x
• Q 2 c, e
• Ex 10B Pg 49-50
• Q3, 5, 7, 8
Thinking Time ?????
• If 3 angles on A are equal to the 3
corresponding angles on the other B, are
the two triangles congruent ?
Ratios and Similar Figures
• Similar figures have corresponding
sides and corresponding angles that
are located at the same place on the
figures.
• Corresponding sides have to have
the same ratios between the two
figures.
Ratios and Similar Figures
A
B
E
F
G
H
Example
C
D
These angles correspond:
These sides correspond:
AB and EF
BD and FH
CD and GH
AC and EG
A and E
B and F
D and H
C and G
Ratios and Similar Figures
14 m
7m
Example
3m
These rectangles
are similar,
because the ratios
of these
corresponding
sides are equal:
7 14

3 6
7 3

14 6
6m
3 6

7 14
14 6

7 3
Proportions and Similar Figures
•A proportion is an equation that states
that two ratios are equivalent.
•Examples:
4 8

n 10
n=5
6 m

3 2
m=4
Proportions and Similar Figures
You can use proportions of
corresponding sides to figure out
unknown lengths of sides of polygons.
16 m
n
10 m
5m
10/16 = 5/n
so n = 8 m
Similar triangles
• Similar triangles are triangles with
the same shape
For two similar triangles,
• corresponding angles have the same measure
• length of corresponding sides have the same
ratio
Example
4
cm
?
25o
12cm
Angle = 90o
2cm
65o
?
Side = 6 cm
Similar Triangles
3 Ways to Prove Triangles
Similar
Similar triangles are like similar
polygons. Their corresponding
angles are CONGRUENT and their
corresponding sides are
PROPORTIONAL.
10
6
3
8
5
4
But you don’t need ALL
that information to be
able to tell that two
triangles are similar….
AA Similarity
• If two angles of a triangle are congruent to
the two corresponding angles of another
triangle, then the triangles are similar.
25 degrees
25 degrees
SSS Similarity
• If all three sides of a triangle are
proportional to the corresponding sides of
another triangle, then the two triangles are
similar.
21
14
18
12
12
8
12 3

8 2
18 3

12 2
21 3

14 2
SSS Similarity Theorem
If the sides of two triangles are in proportion,
then the triangles are similar.
D
A
C
B
F
AB AC BC


DE DF EF
E
SAS Similarity
• If two sides of a triangle are proportional to
two corresponding sides of another triangle
AND the angles between those sides are
congruent, then the triangles are similar.
18
21
14
12
D
A  D
A
AB AC

DE DF
C
B
F
E
SAS Similarity Theorem
If an angle of one triangle is congruent to an
angle of another triangle and the sides including
those angles are in proportion, then the triangles
are similar.
D
A  D
A
AB AC

DE DF
C
B
F
SAS Similarity Theorem
Idea for proof
E
Name Similar Triangles and
Justify Your Answer!
A
D
B
80
80
E
C
ABC ~ ADE by AA ~ Postulate
C
6
D
3
A
10
E
5
B
CDE~ CAB by SAS ~ Theorem
L
5
3
6
K
6
M 6
N
10
O
KLM~ KON by SSS ~ Theorem
20
A
D
30
16
C
24
36
ACB~ DCA by SSS ~ Theorem
B
L
15
P
25
N
9
LNP~ ANL by SAS ~ Theorem
A
Time to work !!!!
oClass work
oHome work
Ex 10C Page 54
Ex 10C Page 54
Q2a to h
Q1a to f
Q3
Q4
Q5
Q7
Q6 a to d
Q9
Q8
Q11
Q10
Q14
Q12
Q15
Q13
Areas of Similar Figures
Activity : Complete the table for each of the
given pairs of similar figures
Conclusion:
If the ratio of the corresponding lengths
of two similar figures is
a
b
then the ratio of their areas is
A1  l1 
 
A 2  l2 
2
a
 
b
2
Thinking Time
Does the identity works for the following
figures ? Why?
A1  l1 
Time to work !!! A   l 
2
 2
•
•
•
•
•
•
•
•
Class work
Ex 10D Pg 62
Q1 a to d
Q3
Q4
Q5
Q8
Q9
•
•
•
•
•
•
•
•
Class work
Ex 10D Pg 62
Q 10
Q12
Q13
Q15
Q16
Q20 - 22
2
Home Work 
Ex 10D Pg 62
• Q2
• Q6
• Q7
• Q11
• Q14
• Q17
• Q18
Volumes of Similar Solids
Activity : Complete the table for each of the
given pairs of similar Solids
Conclusion:
If the ratio of the corresponding lengths
of two similar figures is
a
b
then the ratio of their volumes is
V1  l1 
 
V2  l2 
3
a
 
b
3
Total Surface Area of
similar solids
If the ratio of the corresponding lengths of two similar
figures is
a
b
then the ratio of their total surface areas is
SA1  l1 
 
SA 2  l2 
2
a
 
b
2
Time to work  V1  l1 
3
SA1  l1 
 
 
V2  l2  SA 2  l2 
•
•
•
•
•
•
•
Class work
Ex 10E Pg 67
Q1
Q2
Q3 
Q4
Q5
•
•
•
•
•
Class work
Ex 10E Pg 67
Q6
Q9
Q11
2
Q3 ? How to find the weight of a
similar solid???
• If both solids were made from the same
material,
• Density will be the same 
• Hence using the formula :
Density = Mass  Volume
M 1    V1
and
M 2    V2
M 1   V1

M 2   V2
V1  l1 
 
V2  l2 
3
M 1  l1 
 
M 2  l2 
3
Your favourite moment 
•
•
•
•
•
•
•
•
Home work
Ex 10E Pg 67
Q7
Q8
Q10
Q12
Q13
Q14