Chapter 4 Notes
Section 4-1, Classifying Triangles
Symbol for Triangle:
Classifying Triangles by Angles
Example 1: Classify each triangle as acute, obtuse, equiangular, or right.
Example 2: Classify ∆XYZ as acute, equiangular, obtuse, or right. Explain your reasoning.
Now use the figure to classify ∆YWZ.
Classifying Triangles by Sides
Example 3: Classify each triangle as equilateral, isosceles, or scalene.
8
6
4
Example 4: If Y is the midpoint of VX, and WY= 3.0 units, classify ∆VWY as equilateral, isosceles, or scalene. Explain your
reasoning.
Example 5: If C is the midpoint of BD, classify ∆ABC as equilateral, isosceles, or scalene.
Example 6: Find the measures of the side lengths of this triangle.
9x 1
5x  0.5
4x  1
Example 7: Find the measures of the side lengths of this triangle.
3y  3
2y  5
5 y  19
Assignment: pg. 238; 1-37, 49-52
*Due on Tuesday, November 3*
Section 4-2, Angles of Triangles
Triangle Angle-Sum Theorem
Example 1: The diagram shows the path of the softball in a drill developed by four players. Find the measure of each
numbered angle.
Example 2: Find the measures of each of the numbered angles.
Exterior Angle Theorem
Example 3: Find the value of the variables.
Example 4: Find the values of the variables.
Corollaries
Example 5: Find the measures of the numbered angles.
Assignment: 4-2 Worksheet
*Due on Wednesday, Nov. 4*
Section 4-3a, Congruent Triangles
Two figures are _____________________ if they have the same _______________ and same ____________________.
Corresponding Parts: _____________________________________________________
∆ABC is congruent to ∆XYZ.
Y
B
Z
X
C
A
Corresponding Parts
Sides
Angles
Example 1: ∆BAD is congruent to ∆THE. Name all of the corresponding parts.
A
Sides
H
D
B
E
T
Example 2: ∆QRS is congruent to ∆BRX. Name all of the corresponding parts.
Sides
S
Q
R
Angles
B
X
Angles
Example 3: List all of the corresponding parts.
ABC  FDE
Sides
Angles
Polygon ABCD  Polygon PQRS
Sides
Angles
Example 4: In the diagram, ∆ITP = ∆NGO. Find the values of x and y.
Example 5: In the figure, Quadrilateral JIHK = Quadrilateral QRST. Find a, b, and c.
H 4b o 3a
30o
I
R
J
Q
6
120o
S
c  10
T
K
Example 6: Can you conclude these triangles are congruent? Justify your answer.
Third Angles Theorem
Example 7: Find the missing information in the following proof.
Given: QN  OP, QP  ON
Q  O, NPQ  PNO
Prove: QNP  OPN
Statements
Reasons
1. QN  OP, QP  ON
1. Given
2. NP  NP
2. Reflexive Property of Congruence
3. Q  O
3. Given
NPQ  PNO
4. QNP  ONP
4.
5. QNP  OPN
5. Definition of Congruent
Triangles
Example 8: Write a two-column proof.
Given: L  P, LM  PO, LN  PN , MN  OP
Prove:
LMN  PON
Statements
Reasons
Assignment: Proving Triangles Congruent, Day 1
*Due on Monday, November 9*
Section 4-3b, Triangle Congruence Theorems
The Five Triangle Congruence Theorems
Theorem If ______________ _______________ in one triangle are ________________ to ___________
_____________ in another triangle, the ________________________ must also be ____________________.
xo
85o
xo
Are the triangles congruent?
30o
85o
30o
ASA (Angle, Side, Angle)- If __________ _________ and the __________________ ____________ of one triangle are __________
to ____________ ______________ and the ________________ ____________ of the other, then the triangles are ________.
AAS (Angle, Angle, Side)- If _________ ______________ and a ____________________ _____ of one triangle are __________ to
_______ ___________ and the _________________ _____________________ _________________ of the other triangle, then the
triangles are ______________.
SAS (Side, Angle, Side)- If in two triangles, ___________ ____________ and the _______________ _____________ of one triangle
are _____________ to ____________ ____________ and the ____________________ _____________ of the other, then the
triangles are _______________.
SSS (Side, Side, Side)-
In two triangles, if ___________ ____________ of one triangle are _______________ to
___________ ______________ of the other, then the triangles are ____________.
RHL (Right, Hypotenuse, Leg)- If ______________ ____________ have a ______________ angle and __________
_____________________ are ______________, and another pair of ____________ (___________) are ___________, then the
triangles are ______________.
Example 1: Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in
this lesson?
Example 2: Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in
this lesson?
Example 3: Given the markings on the diagram, is the pair of traingles congruent by one of the congruency theorems in
this lesson?
Example 4: Which theorem, if any, can you use to prove that each pair of triangles is congruent?
Example 5: Which theorem, if any, can you use to prove that each pair triangles are congruent?
Summary: The Five Congruence Theorems
ASA- Pairs of congruent sides contained between two congruent angles
SAS- Pairs of congruent angles contained between two congruent sides
SSS- Three pairs of congruent sides
RHL- A.S.S. condition where matching angles are 90o
AAS- Two angles and the non-included side
You can NOT prove triangles congruent by:
 AAA (Angle Angle Angle)
 RHL (it must be a right triangle, but then it’s called RHL or HL)
Assignment: Ways to Prove Triangles Congruent, Day 2
*Due on Tuesday, Nov. 10*
Triangle Congruence with Proofs
Using CPCTC
CPCTC:
You must:
Assignment: Triangle Congruence with CPCTC
*Due on Thursday, November 12*
Section 4-6, Isosceles and Equilateral Triangles
Practice: Answer the following questions.
U
1. Name the angle opposite of UF.
2. Name the angle opposite of UN.
3. Name the angle opposite of FN.
4. Name the side opposite of Angle U.
5. Name the side opposite of Angle N.
F
N
Isosceles Triangles
Example 1: Name two unmarked congruent angles. Name two unmarked congruent segments.
Example 2: Name two congruent unmarked angles. Name two congruent unmarked sides.
Equilateral Triangles
Example 3: Fill in the blank.
Example 4: Find the requested angle.
mT
Example 5: Find the value of each variable.
Example 6: Find the value of each variable.
Assignment: Worksheet 4-6
*Due on Friday, November 13*