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Geometry Definitions, Postulates, and Theorems
Chapter 4: Congruent Triangles
Section 4.1: Apply Triangle Sum Properties
Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems. 13.0 Students prove relationships between angles in polygons by using properties of
complementary, supplementary, vertical, and exterior angles.
B
A
Triangle –
C
*Classifying Triangles by Sides:
Scalene Triangle –
Isosceles Triangle –
Legs –
Base –
Equilateral Triangle –
*Classifying Triangles by Angles:
Acute Triangle –
Right Triangle –
Legs –
Hypotenuse –
Obtuse Triangle –
Equiangular Triangle –
(over)
Ex. Classify the triangles by their sides and angels.
a)
b)
c)
5
3
120
4
Vertex (plural: vertices) –
B
Adjacent Sides of an Angle –
A
C
Opposite Side from an Angle –
Interior angles –
Exterior angles –
***Theorem 4.1 – Triangle Sum Theorem
B
350
850
C
mB=
A
***Theorem 4.2 – Exterior Angle Theorem
E
670
D
1
F
700
m1=
Corollary To The Triangle Sum Theorem –
Y
X
Z
Ex. A triangle has the given vertices. Graph the triangle
and classify it by its sides. Then determine if it is a
right triangle.
A(5, 4), B(2, 6), C (4,  1)
Ex. Find the value of x and y.
Then classify the triangle by its angles.
Ex. Find the value of x.
Then classify the triangle by its angles.
x0
600
500
x
(2x-18)0
y
Ex. Find the angle measures of the numbered angles.
1
220
2
580
3
200
4
Ex. Find the values of x and y.
y0 x0
680
720
Section 4.2: Apply Congruence and Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
Two geometric figures are congruent if they have exactly the same size and shape, like placing one
figure perfectly onto another figure.
Congruent Figures –
Congruence Statements –
Ex. GIVEN: ABC  DEF
B
F
D
Corresponding Angles – A 
B 
C 
A
E
C
Corresponding Sides –
AB 
BC 
CA 
Ex. Write a congruency statement.
Ex. ABCD  JKHL. Find the value of x and y.
T
L
9 cm
A 910
B
1130
S
Q
4x–3 cm
R
D 86
P
J
(5y–12)0 K
0
C
H
U
***Theorem 4.3 – Third Angle Theorem
Ex.
A
B
F
C
D
E
K
Ex.
G
F
J
I
H
***Theorem 4.4 – Properties of Congruent Triangles
Reflexive Property of Congruent Triangles –
Symmetric Property of Congruent Triangles –
Transitive Property of Congruent Triangles –
Ex. Find the values of x and y.
(8x  2 y)0
(6 x  y)0
29 0
109 0
Section 4.3: Prove Triangles Congruent by SSS
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
***Side-Side-Side (SSS) Congruence Postulate –
B
If Side AB 
E
A
C
D
F
Side BC 
, and
Side CA 
,
then 

by
Ex. Is the congruence statement true?
Explain your reasoning.
WXY  YZW
.
Ex. Is the congruence statement true?
Explain your reasoning.
KJL  MJL
X
Y
L
K
W
J
Z
,
M
Ex. Write a proof.
B
Given: AD  CD , AB  CB
Prove: ABD  CBD
A
D
Structural Support –
Ex. Determine whether the figure is stable. Explain your answer.
a)
b)
c)
C
Section 4.4: Prove Triangles Congruent by SAS and HL
Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are
congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
***Side-Angle-Side (SAS) Congruence Postulate –
B
A
If Side AB 
E
C
D
F
,
Angle A 
, and
Side AC 
,
then 

by
.
Ex. Do you have enough information to prove the triangles are congruent by SAS?
a)
b)
A
Ex. Write a proof.
D
C
Given: AC  EC , DC  BC
Prove: ACB  ECD
B
E
M
Ex. Write a proof.
Given: AB  PB , MB  AP
Prove: MBA  MBP
A
B
P
(over)
Right Triangles: Legs –
Hypotenuse –
***Hypotenuse-Leg (HL) Congruence Theorem –
A
C
D
B
F
E
B
Ex. Write a proof.
Given: AB  BC , BD  AC
Prove: ABD  CBD
A
D
C
Section 4.5: Prove Triangles Congruent by ASA and AAS
Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are
congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
***Angle-Side-Angle (ASA) Congruence Postulate –
B
A
If Angle A 
E
C
D
F
,
Side AC 
, and
Angle C 
,
then 

by
.
***Angle-Angle-Side (AAS) Congruence Postulate –
B
A
If Angle A 
E
C
D
F
,
Angle C 
, and
Side BC 
,
then 
by

.
Ex. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem used.
a)
/
b)
/
c)
/
d)
/
(over)
X
Ex. Write a proof.
Given: WZ bisects XZY and XWY
Prove: WZX  WZY
Z
W
Y
Ex. Write a proof.
Given: C  B , D  F , M is the midpoint of DF
Prove: BDM  CFM
C
B
D
M
F
Section 4.6: Use Congruent Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
***Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.) –
1.
2.
The triangles below are congruent by SAS. Since the triangles are congruent, we know that:
A
X
A  
C  
B
C
Y
Z
Ex. Write a proof.
Given: HJ II LK , JK II HL
Prove: LHJ  JKL
Ex. Write a proof.
AC 
H
J
L
K
M
Given: MS II TR , MS  TR
Prove: A is the midpoint of MT
R
A
S
T
P
Ex. Write a proof.
Given: MP bisects LMN , LM  NM
Prove: LP  NP
L
N
M
Ex. Which triangles can you show are congruent in order to prove the statement? What postulate or
theorem would you use?
b) SW  TY
a) A  C
B
S
X
A
D
C
W
Y
T
Section 4.7: Use Isosceles and Equilateral Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles. 12.0 Students find and use measures of sides and of interior and exterior
angles of triangles and polygons to classify figures and solve problems.
Legs –
Vertex Angle –
Base –
Base Angles –
***Theorem 4.7 – Base Angles Theorem
***Theorem 4.8 – Converse of Base Angles Theorem
Corollary To Base Angles Theorem
Corollary to the Converse of Base Angles Theorem
Section 4.8: Perform Congruence Transformations
Standards: 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations,
translations, and reflections.
Transformation –
Image –
*Three Types of Transformations:
Translation –
Reflection –
Rotation –
Congruence Transformation –
Translate A Figure In The Coordinate Plane –
*Coordinate Notation for a Translation
You can describe a translation by the notation
( x, y )  ( x  a , y  b )
which shows that each point ( x, y ) of a figure is
translated horizontally a units and vertically b units
Ex. Figure ABCD has the vertices
A(4, 2), B(2, 5), C (1,1), and D(3,  1) .
Sketch ABCD and its image after the translation
( x, y )  ( x  5, y  2) .
Reflect A Figure In The Coordinate Plane – The line of reflection is always
the x-axis or the y-axis.
y
*Coordinate Notation for a Reflection
Reflection in the x-axis: ( x, y )  ( x,  y )
Multiply the y-coordinate by -1.
Reflection in the y-axis: ( x, y )  ( x, y )
Multiply the x-coordinate by -1.
x
Ex. Use a reflection in the x-axis to draw the other half
of the figure.
Rotate A Figure In The Coordinate Plane – The center of rotation is the origin.
The direction of rotation can be either clockwise or counterclockwise.
The angle of rotation is formed by rays drawn from the center of rotation through corresponding points
on the original figure and its image.
900 clockwise rotation
600 counterclockwise rotation
y
Ex. Graph PQ and RS . Tell whether RS is a rotation
of PQ about the origin. If so, give the angle and
direction of rotation.
a) P(2, 6), Q(5,1), R(6,  1), S (1,  2)
x
b) P(4, 2), Q(3, 3), R(2, 4), S (3, 3)