SIDES
Scalene – all sides have a
Isosceles – at least two of the
Equilateral – all three sides
different length( measure )
sides have the same measure
have the same measure
11
7
8
5
7
5
4
5
3
ANGLES
Equiangular – all three
angles are equal
Right – one angle is a
Obtuse – one angle is
Acute – all three
right angle and the
other two are acute
obtuse and the other
two are acute
angles are acute
30
51
60
46
72
39
Parts of the Triangle:
B
48
104
- _____ ABC
- Points A, B, and C are
- Two sides sharing a common vertex are adjacent
A
C
- side BC is the side opposite < A
Parts of the Isosceles Triangle:
-
two congruent sides are called
the legs
- 3rd side is the base
use
Parts of the Right Triangle:
- sides that form the right angle are l
- side opposite the right angle is the
Theorem 4.1: TRIANGLE SUM THEOREM
The sum of the measures of the interior angles
of a triangle is _________
Theorem 4.2: EXTERIOR ANGLE THEOREM
The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two
__________________________ angles.
Practice:
Find the measure of the numbered angles:
1.
22
1
2
3
50
58
7
1
2
6
125
3
4
20
2.
4
5
85
3. Solve for x:
(2x+3)o
(4x+8)o
51o
4. Solve for x: Find the angle measures and classify the triangle by its angles.
m  A = (3x – 17)
m  B = ( x + 40)
m  C = ( 2x – 5)
8
9
65
Definition of Congruence:
Two geometric figures are congruent if they have exactly the same _______________
and same ______________.
There is a correspondence between their angles and sides such that
1.) corresponding angles are congruent
2.) corresponding sides are congruent
ORDER OF THE
LETTERS IS
IMPORTANT!!!
E
B
∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭
A
C
Corresponding Angles
Congruent Figures
All the parts of
one figure are
congruent to the
corresponding
parts of the other.
D
F
Corresponding Sides
Theorem 4.3: Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then
_______________________________________________
Example:
If  A and  B of ∆ABC are congruent to  D and  E of
then _______ ≅ ______.
Reflexive property of congruent triangles:
∆ABC
Symmetric property of congruent triangles: If
∆DEF respectively,
≅ ___________
∆ABC
≅
∆DEF
then _______________
Transitive property of congruent triangles:
If
∆ABC
≅
∆DEF
and
∆DEF
≅
∆JKL
then ____________________
Y
Recall: Isosceles triangle
(Label as much as you can)
Z
X
A
Theorem 4.7 Base Angle Theorem
If two sides of a triangle are congruent, then the opposite
angles are ____________________
If AB ≅ AC, then __________________
A
Theorem 4.8 Converse of Base Angle Theorem
If two angles of a triangle are congruent, then the sides opposite
them are _____________________
If  B ≅  C, then ______________
C
B
B
C
Since the converse of the Base Angle Theorem is true what is the biconditional?
Examples:
1.) In isosceles ∆PQR. With base QR, PQ = 2x+3 and PR = 9x–11. What is the value of x?
Equilateral and Equiangular
If a triangle is _______________, then it is ___________________.
If a triangle is _______________, then it is ___________________.
Bicondictional: A triangle is ______________ iff it is __________________.
Name: __________________
1.)
(a)
∆ABC ≅ ∆DEF
Date: _________
B
E
80
 A ≅ ________
∆FDE ≅ ________
2.) Given:
A
35o
A
(b) FD ≅ ________
(c)
Block: _____
C
(d) BA = ________
o
8 cm
F
D
(e) mLA = mL_____ = _____
∆ABC ≅ ∆DEF, find the values of x and y.
F
87o
B
E
3y
42o
(5x + 2)o
D
C
3.) Identify corresponding angles and sides.
Corresponding Angles
J
Corresponding Sides
F
G
K
H
4. Given:  L   X and  J   Y. Find the value of k.
(3k -20)o
J 35o
L
M
Y
X
Z
H
5.) (a) If HG ≅ HK, then _______ ≅ _______
(b) If  KHJ ≅  KJH, then _______ ≅ _______
G
J
K
Find the unknown measure.
6.)
7.)
8.)
P
?
15
R
Find the values of x and y.
9.)
10.)
11.) Find the perimeter of the triangle.
3
18
Q
Postulate 19 Side-Side-Side (SSS) Congruence Postulate
R
If three sides of one triangle are congruent to _______ _________
of a second triangle, then they two triangles are congruent.
If
Side
Side
Side
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝑅𝑆,
̅̅̅̅
̅̅
̅̅ , and
𝐵𝐶 ≅ 𝑆𝑇
̅̅̅̅ ≅ 𝑇𝑅
̅̅̅̅,
𝐶𝐴
B
C
S
then
∆ _________ ≅ ∆_________.
A
T
V
Postulate 20 Side-Angle-Side (SAS) Congruence Postulate
If two sides and the ____________ angle of one triangle are
congruent to two sides and the _____________ angle of a
second triangle, then they two triangles are congruent.
If
Side
Angle
Side
̅̅̅̅
̅̅̅̅,
𝑅𝑆 ≅ 𝑈𝑉
∠𝑅 ≅ ∠𝑈, and
̅̅̅̅ ≅ ̅̅̅̅̅
𝑅𝑇
𝑈𝑊 ,
then
R
U
T
∆ _________ ≅ ∆_________.
W
S
Included Angle: The angle ___________________ two sides.
Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate
E
B
If two angles and the included side of one triangle are
congruent to two angles and the included side of a second
triangle, then they two triangles are congruent.
If
Angle
Side
Angle
∠𝐴 ≅ ∠𝐷,
̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐷𝐹 , and
∠𝐶 ≅ ∠𝐹,
A
then
∆ _________ ≅ ∆_________.
Included Side: The side ___________________ two angles.
D
C
F
Theorem 4.6 Angle-Angle-Side (AAS) Congruence Postulate
E
If two angles and a _____ - ____________ side of one triangle are congruent
to two angles and the corresponding _____ - ____________ side of a second
triangle, then they two triangles are congruent.
B
D
If
Angle ∠𝐴 ≅ ∠𝐷,
Angle ∠𝐶 ≅ ∠𝐹, and
then ∆ _________ ≅ ∆_________.
̅̅̅̅
̅̅̅̅
Side
𝐵𝐶 ≅ 𝐸𝐹 ,
A
Theorem 4.5 Hypotenuse-Leg (HL) Congruence Theorem
F
C
D
A
If the hypotenuse and a leg of a _______ triangle are congruent
to the hypotenuse and a leg of a second right triangle,
then the two triangles are congruent.
If
F
̅̅̅̅
Hypotenuse
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐸 ,
̅̅̅̅ ≅ 𝐷𝐹
̅̅̅̅ , and
Leg
𝐴𝐶
o
𝑚∠𝐶 = 𝑚∠F = 90
C
then
E
B
∆ _________ ≅ ∆_________.
Determine whether the congruence statement is true.
Explain your reasoning (SSS, SAS, ASA, AAS, HL)
1.)
2.) ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶
∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
3.) ∆𝐽𝐾𝐿 ≅ ∆𝐿𝑀𝑁
C
B
J
D
B
L
N
C
A
E
A
D
4.) ∆𝑌𝑋𝑍 ≅ ∆𝑊𝑋𝑍
K
5.) ∆𝐽𝐾𝐿 ≅ ∆𝑅𝑆𝑇
Z
6.) ∆𝑄𝑅𝑆 ≅ ∆𝑇𝑉𝑆
S
S
R
X
Y
R
Q
L
J
W
M
K
V
T
7.) Indicate the additional information needed to enable us to apply the specified congruence
postulate.
For ASA:
For SAS:
For AAS:
T
CPCTC – Corresponding
Parts of Congruent
Triangles are Congruent
By definition, congruent triangles have congruent corresponding parts. If you can prove that two triangles
are congruent, you know that their corresponding parts must be congruent as well.
1.) Given: A is the midpoint of MT and SR
Prove: MS TR
R
M
A
T
S
You cannot use
CPCTC until you prove
the triangles are
congruent.
2.)
STATEMENTS
REASONS
1. A is the midpoint of MT
1.
2. MA  TA
2.
3. A is the midpoint of SR
3.
4. SA  RA
4.
5.
MAS  TAR
5.
6.
MAS  TAR
6.
7.
M T
7.
8. MS TR
8.
Given: ̅̅̅̅
𝐴𝐵 and ̅̅̅̅
𝐹𝐷 bisect each other
Prove: ∠𝐴𝐷𝐹 ≅ ∠𝐹
Statements
Reasons
3.)
Given: ̅̅̅̅̅
𝑊𝑌 bisects ∠𝑍𝑊𝑋
∠𝑍 ≅ ∠𝑋
Prove: ̅̅̅̅
𝑍𝑌 ≅ ̅̅̅̅
𝑋𝑌
Statements
4.)
Reasons
̅̅̅̅
Given: ̅̅̅̅
𝐷𝐸 is perpendicular to 𝐹𝐶
̅̅̅̅
̅̅̅̅
𝐷𝐹 ≅ 𝐷𝐶
Prove: ∠𝐹 ≅ ∠𝐶
Statements
Reasons