Chapter 5
Applying Congruent Triangles
Warm Up For
Chapter 5
5.2 Right
Triangles
Internet
Activity

Objective: Identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle
How will I use this?
Special segments are used in triangles to solve problems involving engineering, sports and physics.
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Me!!
Chapter 5
Median
Perpendicular
Bisector
Altitude
An example to tie it all together
Angle
Bisector

A segment that connects a vertex of a triangle to the midpoint of the side opposite the vertex.

A line segment with 1 endpoint at a vertex of a triangle and the other on the line opposite that vertex so that the line segment is perpendicular to the side of the triangle.

Perpendicular Bisector:
A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.

Theorem 5.1: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
Theorem 5.2: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
Chapter 5
Median
Perpendicular
Bisector
Altitude
Angle
Bisector

Theorem 5.3: Any point on the bisector of an angle is equidistant from the sides of the angle.
Theorem 5.4: Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle.
Chapter 5
Median
Perpendicular
Bisector
Altitude
Angle
Bisector

      and m  C  5 x  10
Find the value of x and the measure of each angle.
 ,

m  A  45
x  20
m  B  45
m  C  90
How did I get that? Click the answer to see!
BONUS!!!
What type of triangle is ABC?
Click me to find the Answer!!
Section
5.1

BONUS!!!
What type of triangle is ABC?
Click me to find the Answer!!

Because the question give you angle measures, we take the sum of the angles and set them equal to 180.
Chapter 5
2 x  5 m  A
 3 x  15 m  B
 5 x  10 m  C
10 x  20  180
Combine like terms!
 180
10 x  200
Add 20 to both sides!
m  A
x  20
Divide by 10 on both sides!
m  B
m  C
Section 5.1

Use substitution for the answer you found for it into the equation for angle A. x and plug m  A  2 x  5
x  20
m  A  2 ( 20 )  5


40  5
45
Chapter 5
m  B
m  C
Section 5.1

Use substitution for the answer you found for it into the equation for angle B. x and plug m  B  3 x  15
x  20
m  B  3 ( 20 ) 
15


60  15
45
Chapter 5
m  A
m  C
Section 5.1

Use substitution for the answer you found for it into the equation for angle C. x and plug m  C  5 x  10
x  20
m  C  5 ( 20 ) 
10
 100  10
 90
Chapter 5
m  A
m  B
Section 5.1

Triangle ABC is a right isosceles triangle
Chapter 5
Section 5.1

What is an Angle
Bisector?
Click me to find out!
Move my vertices around and see what happens!!
Angle
Bisector
Theorems
Example
Section
5.1

Draw the three medians of triangle ABC.
Name each of them.
B
Answer
C
A

Draw the three medians of triangle ABC.
Name each of them.
B
F D
A
E
C
Back to Section
5.1

Draw the three altitudes, QU, SV, and RT.
R
Answer
S
Q

Draw the three altitudes, QU, SV, and RT.
V
U
R
Q
T
S
Back to Section
5.1

Draw the three lines that are perpendicular bisectors of XYZ.
Y
X
Label the lines l
, m
, and
Z n
.

Draw the three lines that are perpendicular bisectors of XYZ.
Y m l
X n Z
Back to Section
5.1

 and the measure of AC.
B
3 x  2 x  6
A
2 x  3
D
3 x  2
C

A
 and the measure of AC.
B
3 x  2 x  6 x
AC


4
21
2 x  3
D
3 x  2
C
Show me how you got those answers!
Back to Section
5.1

 and the measure of AC.
B Means that the angle is split into 2 congruent parts. Set the two angles equal to each other and solve.
3 x  2 x  6
Once you find x, plug it into
AD and DC. Since you are looking for the total length,
AC, use segment addition to find the total length.
A
2 x  3
D
3 x  2
C
Back to Section
5.1

Given:  RAB is isosceles with vertex angle RAB
EA is the bisector of  RAB
Prove: EA is a median
1
 RAB is isosceles
2
RA  BA
1
EA is the bisector of  RAB
4
AE  AE
3
 RAE   BAE
5
 RAE   BAE
6
RE  BE
7
AE is a median
1. Given
2. Def of Isos Triangle
3. Def of Angle Bisector
4. Reflexive
5. SAS
6. CPCTC
7. Def of Median
5.1 Proofs

Given: 
STU is equilatera l
TW is an angle bisector of

STU
Prove: TW is a median of

STU
1

STU is equilatera l
2
TU

TS
1
TW is angle bisector of

STU
4
TW

TW
3

UTW
 
STW
5

UTW
 
STW
6
UW

SW
7
T W is a median
1. Given
2. Def of Equilateral Triangle
3. Def of Angle Bisector
4. Reflexive
5. SAS
6. CPCTC
7. Def of Median
We’re done, take me back to the beginning!


SGB has vertices
S(4,7)
G(6,2) and
B(12,-1)
Keep clicking to see graph!
Determine the coordinate s of point J on GB so that
SJ is the median of

SGB
Midpoint
See the Work!!
S
G
B


SGB has vertices S(4,7), G(6,2) and B(12,-1)
What is the
Midpoint
Formula?
Midpoint of
6

12
,
2
2
 
1
2

9 ,
1
2
GB x
2
 x
1 ,
2 y
2
 y
1
2
Next Question

Point M has coordinate s (8,3).
Is GM an altitude of

SGB ?
GM must be

to SB
Slope of
Slope of
GM
SB


3

8
1
12




2
6
7
4


1
2

1
What can we conclude?
We’re done, take me back to the beginning!

An Internet Activity
CLICK TO BEGIN

Take notes as you read along with each Theorem or
Postulate!!
Leg Leg
Theorem
Hypotenuse
Angle
Theorem
Leg Angle
Theorem
Click on the triangle and learn about the
Theorems or
Postulates.
Hypotenuse
Leg
Postulate
Click me when done

I finished!
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Solving for variables
Stating additional information

Example
1
Example 3
Example
2

Example
1
Example
2
Example
3

Find the value of x and y so that

DEF
 
PQR by HA.
D P
2 x

3
4 x

1
F 
3 y

10


E Q 
4 y

20


Answer
R

3 y

10

4 y

20 y

30
2 x

3

4 x

1

2 x
 
2 x

1

23
Find the value of x and y so that

DEF
 
PQR by LL.
E
56
F
D
2 x

4
Q
2 y

9
R
P
Answer

23

2 y

9
32

2 y
16
 y
56

2 x

4
52

2 x
26
 x

Find the value of x and y so that

DEF
 
PQR
 y

3

 by
E
F
P
LA.
6 2 x

4
D
R 
3 y

20


Q
Answer

3 y

20
 y

50
2 y

30 y

15
2 x

4

6
2 x

10 x

5
Back to
Beginning

State the additional information needed to prove the pair of triangles congruent by LA.
J
M
L
K
Answer

Proving triangles congruent by LA means a leg and an angle of the right triangle must be congruent.
J
M
LM

M

LK
 
K
K
L
OR JL

JL

MJL
 
KJL
Next Example

State the additional information needed to prove the pair of triangles congruent by HA .
S
T
Z
V
X Y
Answer

State the additional information needed to prove the pair of triangles congruent by HA .
S
V X
T
Z
Y
The keyword was additional . When proving triangles congruent by HA, all that is needed is to show that the hypotenuse is congruent on each triangle as well as an acute angle. In these triangles both are already shown so there is no
ADDITIONAL information needed.
Next Example

State the additional information needed to prove the pair of triangles congruent by LA.
B
C
D
A
F
Answer

State the additional information needed to prove the pair of triangles congruent by LA.
B
C 
C
 
F
A
D
BC

DF
OR

BAC
 
FAD
BA

AD
F
Back to
Beginning