Extra Practice

Report
Chapter 5
Applying Congruent Triangles
Warm Up For
Chapter 5
5.2 Right
Triangles
Internet
Activity
5.1 Special Segments in Triangles
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.
Chapter 5
Median
Perpendicular
Bisector
Altitude
An example to tie it all together
Click
Me!!
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.
Theorems
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
Theorems
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
Warm UP
In  ABC , m  A  2 x  5 , m  B  3 x  15 ,
and m  C  5 x  10
Find the value of x and the measure of
each angle.
Warm Up Answers
x  20
m  A  45
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.
}
}
}
2 x  5  3 x  15  5 x  10  180
mB
mA
mC
10 x  20  180
Combine like terms!
10 x  200
Add 20 to both sides!
x  20
Divide by 10 on both
sides!
Chapter 5
mA
mB
mC
Section 5.1
Use substitution for the answer you found for x and plug
it into the equation for angle A.
m  A  2x  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 x and plug
it into the equation for angle B.
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 x and plug
it into the equation for angle C.
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
Why is that??
Chapter 5
Section 5.1
Angle Bisector
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
Median Example
Draw the three medians of triangle ABC.
Name each of them.
B
Answer
A
C
Median Example
Draw the three medians of triangle ABC.
Name each of them.
B
F
A
D
E
C
Back to Section
5.1
Altitude Example
Draw the three altitudes, QU, SV, and RT.
R
Answer
Q
S
Altitude Example
Draw the three altitudes, QU, SV, and RT.
V
U
Q
R
T
S
Back to Section
5.1
Perpendicular Bisector Example
Draw the three lines that are perpendicular
bisectors of XYZ.
Y
Answer
X
Label the lines l,
Z
m, and n.
Perpendicular Bisector Example
Draw the three lines that are perpendicular
bisectors of XYZ.
Y
m
X
l
n
Z
Back to Section
5.1
Angle Bisector Example
If BD bisects  ABC, find the value of x
and the measure of AC.
B
3x  2
A
x  6
2x  3
D
3x  2
C
Answer
Angle Bisector Example
If BD bisects  ABC, find the value of x
and the measure of AC.
B
3x  2
x  4
x  6
AC  21
Show me how
you got those
answers!
A
2x  3
D
3x  2
C
Back to Section
5.1
Angle Bisector Example
If BD bisects  ABC, find the value of x
and the measure of AC.
Means that the angle is
split into 2 congruent
parts. Set the two angles
equal to each other and
solve.
A
B
3x  2
x  6
2x  3
D
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.
3x  2
C
Back to Section
5.1
5.1 Proofs
Together
YOU TRY!!!
Given:
 RAB is isosceles
with vertex angle RAB
EA is the bisector
of  RAB
Prove: EA is a median
2
1
 RAB is isosceles
RA  BA
1
5
3
EA is the bisector
of  RAB
 RAE
 RAE   BAE
  BAE
6
4
RE  BE
AE  AE
7
1. Given
2. Def of Isos Triangle
3. Def of Angle Bisector
4. Reflexive
5. SAS
6. CPCTC
7. Def of Median
AE is a median
5.1 Proofs
Given: STU is equilatera l
TW is an angle bisector of  STU
Prove: TW is a median of  STU
2
1
 STU is equilatera
l
TU  TS
1
5
 UTW   STW
3
TW is angle bisector
of  STU
 UTW
  STW
6
4
UW  SW
TW  TW
7
5. SAS
1. Given
2. Def of Equilateral Triangle 6. CPCTC
7. Def of Median
3. Def of Angle Bisector
4. Reflexive
T W is a median
We’re done, take me back
to the beginning!
Example
 SGB has vertices
S(4,7)
G(6,2) and
B(12,-1)
Determine
Keep
clicking to
see graph!
the coordinate
s
of point J on GB so that
median of  SGB
SJ is the median
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 GB
 6  12 2   1 
,


2 
 2
 1
  9, 
 2
 x 2  x1 y 2  y 1 
,


2
2


Next Question
Point M has coordinate
Is GM an altitude
s (8,3).
of  SGB ?
GM must be  to SB
Slope of GM 
Slope of SB 
32
86
1 7
12  4

1
2
What can we conclude?
 1
We’re done, take me back to
the beginning!
5.2 Right Triangles
An Internet Activity
CLICK TO BEGIN
Take notes as you
read along with
each Theorem or
Postulate!!
Leg Leg
Theorem
Click on the
triangle and
learn about the
Theorems or
Postulates.
Hypotenuse
Angle
Theorem
Leg Angle
Theorem
Hypotenuse
Leg
Postulate
Click me
when done
Examples
I finished!
Click me!!
Solving for
variables
Stating additional
information
Solve for…
Example
1
Example 3
Example
2
State the additional information.
Example
Example
Example
1
2
3
Find the value of x and y so that
 DEF   PQR by HA.
P
D
2x  3
F
3 y  10 
4x 1

E
Q
4 y  20 
Answer

R
3 y  10  4 y  20
y  30
2x  3  4x 1
 2 x  2
x 1
Find the value of x and y so that
 DEF   PQR by LL.
E
56
23
F
D
2x  4
Q
2y 9
R
P
Answer
23  2 y  9
56  2 x  4
32  2 y
52  2 x
16  y
26  x
Find the value of x and y so that
 DEF   PQR by LA.
 y  3
E

FP
2x  4
6
D
R
3 y  20 

Q
Answer
3 y  20  y  50
2 y  30
y  15
2x  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
K
L
Answer
Proving triangles congruent by LA
means a leg and an angle of the right
triangle must be congruent.
J
M
LM  LK
M  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
V
T
X
Z
Y
Answer
State the additional information needed
to prove the pair of triangles congruent
by HA.
S
V
T
X
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
A
C
D
F
Answer
State the additional information
needed to prove the pair of triangles
congruent by LA.
B
C
C  F
BC  DF
A
D
OR
 BAC   FAD
BA  AD
F
Back to
Beginning

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