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Chapter 4
Congruent Triangles
4.1
Triangles and Angles
Parts of Triangles

Vertex


Points joining the sides of a triangle
Adjacent Sides

Sides that share a common vertex
Classification by Sides

Equilateral


Isosceles


3 congruent sides
At least 2 congruent sides
Scalene

No congruent sides
Classification by Angles

Acute


Equiangular


3 congruent angles
Right


3 acute angles
1 right angle
Obtuse

1 obtuse angle
Parts of Isosceles Triangles

Legs


The sides that are congruent.
Base

The non-congruent side.
Isosceles triangles

Base angles are congruent.
Vertex angle
Base angles
Base
legs
Parts of Right Triangles

Hypotenuse


The side that is opposite the right angle.
It is always the longest side.
Legs

The sides that form the right angle
Right Triangles
hypotenuse
leg
leg
Interior Angles

The angles on the inside of a triangle.
Triangle Sum Conjecture

The sum of the measures of the angles
in every triangle is 180.
Example
Find the measure of each angle.
2x + 10
x
x+2
Exterior Angles

The angles that are adjacent to the interior
angles

The exterior angles always add to equal 360°
Definitions
Exterior Angle
Adjacent
Interior
Angle
Remote Interior Angles
Exterior Angles of a Triangle
Use your straightedge to draw a
triangle.
 Extend one side out as shown

B
b
a
A
c x
C
Exterior Angles of a Triangle
Trace angles a and b onto a
transparency so that they are adjacent.
 How does this compare to angle x?

B
b
a
a
b
A
c x
C
Triangle Exterior Angle
Conjecture

The measure of the exterior angle of a
triangle is equal to the sum of the
measures of the remote interior angles
B
b
a
A
c x=a+b
C
Example
Find the missing measures
80°
53°
Example
Find the missing measures
60°
120°
Example

Page 199 #37
(2x – 8)°
x°
31°
4.2
Congruence and Triangles
Terms

Congruent


Corresponding angles


Figures that are exactly the same size and shape
are congruent
The angles that are in corresponding positions are
congruent
Corresponding sides

The sides that are in corresponding positions are
congruent
Naming Congruent Figures

When a congruence statement is made
it is important to match up
corresponding parts.
Third Angle Theorem

If two angles in one triangle are equal
to two angles in another triangle, then
the third angles in each triangle are also
equal.
Examples 1

What is the measure
of:
(page 205)
P
ΔLMN  ΔPQR
Q
N
M
P
R
45°
N
Which side is
congruent to
105°
L
M
R
segment QR
Segment LN
Example 2

Given ABC  PQR, find the values of
x and y.
R
(6y – 4)°
Q
A
85°
B 50°
P (10x + 5)°
C
4.3
Proving Triangles
Congruent SSS and SAS
Warm-Up
Complete the following statement
B
BIG 
A
I
R
G
T
Definitions

included angle


An angle that is between two given sides.
included side

A side that is between two given angles.
Example 1

Use the diagram.
Name the included
angle between the
pair of given sides.
JK and KL
PKand LK
KPand PL
J
K
L
P
Triangle Congruence Shortcut

SSS

If the three sides of one triangle are
congruent to the three sides of another
triangle, then the triangles are congruent.
Triangle Congruence Shortcuts

SAS

If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then the
triangles are congruent.
Example 2
Complete the congruence statement.
 Name the congruence shortcut used.
U

S
T
V
W
STW  
Example 3
Determine if the following are
congruent.
 Name the congruence shortcut used.
H
L

I
M
N
HIJ  LMN
J
Example 4
Complete the congruence statement.
 Name the congruence shortcut used.
A
B

C
X
O
XBO  
R
Example 5
Complete the congruence statement.
 Name the congruence shortcut used.
SPQ  

P
T
S
Q
4.4
Proving Triangles
Congruent ASA and AAS
Triangle Congruence Shortcuts

ASA

If two angles and the included side of one
triangle are congruent to two angles and
the included side of another triangle, then
the triangles are congruent.
Triangle Congruence Shortcuts

SAA

If two angles and a non-included side of
one triangle are congruent to the
corresponding angles and side of another
triangle, then the two triangles are
congruent.
Example 1
Complete the congruence statement.
 Name the congruence shortcut used.

Q
U
QUA  
D
A
Example 2
Complete the congruence statement.
 Name the congruence shortcut used.
M
R
N

Q
P
RMQ  
Example 3
Determine if the following are
congruent.
 Name the congruence shortcut used.

F
B
E
A
ABC  FED
C
D
4.6
Isosceles, Equilateral, and
Right Triangles
Warm-Up 1
Find the measure of each angle.
60°
a
90°
90°
30°
b
Warm-Up 2
Find the measure of each angle.
110
90
150
Isosceles triangles
The base angles of an isosceles triangle
are congruent.
 If a triangle has at least two congruent
angles, then it is an isosceles triangle.


If the sides are congruent then the base
angles are congruent.
Example 1
35°
x
Example 2
b
15°
a
Example 3
Find each missing measure
m
n
10 cm
63°
p
Equilateral Triangles
If a triangle is equilateral, then it is
equiangular.
 If a triangle is equiangular, then it is
equilateral.

Hypotenuse-Leg (HL)

If the hypotenuse and the leg of a right
triangle are congruent to the
hypotenuse and leg of a second right
triangle, then the two triangles are
congruent.
Example 4

Find the value of x
2x in
12 in
Example 5

Find the value of x and y.
y
x
Example 6

Find the value of x and y.
x°
75°
x°
y°
Chapter 8
Similarity
8.1
Ratio and Proportion
Ratios

The ratio of a to b can be written as
a/b
 a : b


The denominator cannot be zero
Simplifying Ratios

Ratios should be expressed in simplified form


6:8 = 3:4
Before reducing, make sure that the units are
the same.

12in : 3 ft
12in : 36 in
1: 3
Examples (page 461)

Simplify each ratio
10.
16 students
24 students
12.
22 feet
52 feet
18.
60 cm
1m
Examples (page 461)

Simplify each ratio
20.
2 mi
3000 ft
24.
20 oz.
4 lb
There are 5280 ft in 1 mi.
There are 16 oz in 1 lb.
Examples (page 461)

Find the width to length ratio
14.
16 mm
20 mm
16.
12 in.
2 ft
Using Ratios Example 1

The perimeter of the isosceles triangle
shown is 56 in. The ratio of LM : MN is
5:4. Find the length of the sides and
the base of the triangle.
L
N
M
Using Ratios Example 2

The measures of the angles in a
triangle are in the extended ratio 3:4:8.
Find the measures of the angles
4x
8x
3x
Using Ratios Example 3

The ratios of the side lengths of ΔQRS
to the corresponding side lengths of
ΔVTU are 3:2. Find the unknown
lengths.
Q
U
T
2 cm
V
S
18 cm
R
Proportions

Proportion
Ratio = Ratio
 Fraction = Fraction


Solving Proportions


Cross multiply
Let the means equal the extremes
Properties of Proportions

Cross Product Property
a c
If  , then ad  bc
b d

Reciprocal Property
a c
b d
If  , then 
b d
a c
Solving Proportions Example 1
9 6

14 x
Solving Proportions Example 2
s5 s

4
10
Solving Proportions Example 3
A photo of a building
has the
measurements
shown. The actual
building is 26 ¼ ft
wide. How tall is
it?
2.75 in
1 7/8 in
8.2
Problem solving in
Geometry with Proportions
Properties of Proportions
a c
a b
If  , then 
b d
c d
a c
ab cd
If  , then

b d
b
d
Example 1

Tell whether the statement is true or
false

A.
s 15
s 3
If
 , then 
10 t
t 2

B.
3 5
3 x 5 y
If  , then

x y
x
y
Example 2

In the diagram MQ LQ

MN LP
Find the length of LQ.
M
6
N
15
13
Q
L
5 P
Geometric Mean

Geometric Mean

The geometric mean between two
numbers a and b is the positive number x
such that
a x

x b
Example 3

Find the geometric mean between 35
and 175.
Example 4

You are building a scale model of your
uncle’s fishing boat. The boat is 62 ft
long and 23 ft wide. The model will be
14 in. long. How wide should it be?
8.3
Similar Polygons
Similar Polygons
Polygons are similar if and only if
the corresponding angles are congruent
and
 the corresponding sides are
proportional.



Similar figures are dilations of each
other. (They are reduced or enlarged
by a scale factor.)
The symbol for similar is 
Example 1
Determine if the sides of the polygon are
proportional.
8m
12 m
6m
8m
6m
Example 2
Determine if the sides of the polygon
are proportional.
15 m
9m
5m
3m
12 m
4m
Example 3
Find the missing measurements.
HAPIE  NWYRS
6
H
A
P
5
AP =
EI =
SN =
YR =
E
I
4
18
W
24
Y
N
S
21
R
Example 4
Find the missing measurements.
QUAD  SIML
S 12
L
65º
A
D
8
125º
Q 20
25
I 95º
U
M
QD =
MI =
mD =
mU =
mA =
8.4/8.5
Similar Triangles
Similar Triangles

To be similar, corresponding sides
must be proportional and
corresponding angles are congruent.
Similarity Shortcuts
AA Similarity Shortcut
If two angles in one triangle are
congruent to two angles in another
triangle, then the triangles are similar.
Similarity Shortcuts
SSS Similarity Shortcut
If three sides in one triangle are
proportional to the three sides in
another triangle, then the triangles are
similar.
Similarity Shortcuts
SAS Similarity Shortcut
If two sides of one triangle are
proportional to two sides of another
triangle and
their included angles are congruent,
then the triangles are similar.
Similarity Shortcuts
We have three shortcuts:
AA
SAS
SSS
Example 1
9
g
6
4
7
10.5
Example 2
k
32
h
50
24
30
Example 3
36
42
m
24
1. A flagpole 4 meters tall casts a 6 meter
shadow. At the same time of day, a nearby
building casts a 24 meter shadow. How tall is
the building?
4
m
6m
24m
2. Five foot tall Melody casts an 84 inch
shadow. How tall is her friend if, at the same
time of day, his shadow is 1 foot shorter than
hers?
3. A 10 meter rope from the top of a flagpole
reaches to the end of the flagpole’s 6 meter
shadow. How tall is the nearby football
goalpost if, at the same moment, it has a
shadow of 4 meters?
10m
6m
4m
4. Private eye Samantha Diamond places a
mirror on the ground between herself and an
apartment building and stands so that when
she looks into the mirror, she sees into a
window. The mirror is 1.22 meters from her
feet and 7.32 meters from the base of the
building. Sam’s eye is 1.82 meters above the
ground. How high is the window?
1.82
1.22
7.32
8.6
Proportions and Similar
Triangles
Proportions

Using similar triangles missing sides
can be found by setting up proportions.
Theorem

Triangle Proportionality Theorem

If a line parallel to one side of a triangle
intersects the other two sides, then it
divides the two sides proportionally.
Q
T
RT RU
If TU || QS ,then

.
TQ US
R
S
U
Theorem

Converse of the Triangle Proportionality
Theorem

Q
If a line divides two sides of a triangle
proportionally, then it is parallel to the third
side.
RT RU
T
If
TQ
R
S
U

US
, thenTU || QS .
Example 1

In the diagram, segment UY is parallel
to segment VX, UV = 3, UW = 18 and
XW = 16. What is the length of
segment YX?
U
V
W
Y
X
Example 2

Given the diagram, determine whether
segment PQ is parallel to segment TR.
Q
9.75
P
R
9
26
T
24
S
Theorem

If three parallel lines intersect two
transversals, then they divide the
transversals proportionally.
Theorem

If a ray bisects an angle of a triangle,
then it divides the opposite side into
segments whose lengths are
proportional to the lengths of the other
two sides.
Example 3

In the diagram, 1  2  3, AB =6,
BC=9, EF=8. What is x?
C
9
B
6
A
1
D
3
2
8
x
E
F
Example 4

In the diagram, LKM  MKN. Use
the given side lengths to find the length
of segment MN.
15
L
N
M
3
17
K
5. Juanita, who is 1.82 meters tall, wants
to find the height of a tree in her
backyard. From the tree’s base, she
walks 12.20 meters along the tree’s
shadow to a position where the end of
her shadow exactly overlaps the end of
the tree’s shadow. She is now 6.10
meters from the end of the shadows.
How tall is the tree?
6.10
1.82
12.20