Download Section 4-1 Classifying Triangles

Pre-AP Geometry Notes
Name: ______________________________
Section 4-1 Classifying Triangles
Objectives: Identify and classify triangles by angle measures. Identify and classify angles by side
measures.
Example 1:
right.
Use the best description to classify each triangle: acute, equiangular, obtuse, or
b. EFH
a. EHG
H
c. HFG
30°
60°
60°
30°
E
F
Example 2:
scalene.
G
Use the best description to classify each triangle: equilateral, isosceles, or
15
B
b. ABD
a. ABC
A
c. ACD
18
15
C
5
D
You can also use the properties of isosceles and equilateral triangles to find missing values.
Example 3: Find the measures of the sides of each triangle.
a. isosceles triangle ABC
b. equilateral triangle FGH
9x-1
G
B
A
2y+5
5x-0.5
4x+1
C
3y-3
F
5y-19
H
Pre-AP Geometry Notes
Name: ______________________________
Section 4-2 Angles of Triangles
Objectives: Apply the Triangle Angle Sum Theorem. Apply the Exterior Angle Theorem. Apply the Third
Angle Theorem.
Auxiliary Line –
Triangle Sum Theorem – The sum of the interior angle measures of a triangle is 180 .
B
Let’s prove this theorem…
Given: ABC
2
Prove: m1  m2  m3  180
Statements:
1
A
Reasons:
3
C
Exterior Angle of a Triangle –
Remote Interior Angle of a Triangle –
Exterior Angle Theorem – The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two remote interior angles.
E
2
Let’s prove this theorem…
Given: DEF with exterior angle 4
Prove: m1  m2  m4
Statements:
Reasons:
D
1
3
4
F
G
Triangle Angle-Sum Corollaries 

The acute angles of a right triangle are complementary.
There can be at most one right or obtuse angle in a triangle.
Example 1:
a.
Find the measures of each numbered angle.
b.
c.
Example 2:
Find the measure of JKL in the triangle shown.
Pre-AP Geometry Notes
Name: ______________________________
Section 4-3 Congruent Triangles
Objectives: Name and use corresponding parts of congruent polygons. Prove triangles congruent
using the definition of congruence.
Example 1: Show that the polygons are congruent by identifying all the congruent
corresponding parts. Then write a congruence statement.
Example 2: Use the fact that corresponding parts of congruent triangles are congruent
(CPCTC) to find the values of x and y .
ABC  DFE
Third Angle Theorem – If two angles of one triangle are congruent to two angles of another
triangle, then the third pair of angles are congruent.
S
D
Let’s prove this theorem…
Given: R  B ; D  S
Prove: E  H
Statements:
Example 3:
Reasons:
R
Complete the following proof.
Given: J  P , JK  PM , JL  PL , and L is the midpoint of KM .
Prove: JLK  PLM
M
J
L
P
K
Statements:
Reasons:
E
B
H
PreAP Geometry Notes
Name______________________
Sections 4-4 & 4-5 Proving Triangles Congruent Date ________
Objectives: Complete the Triangle Congruence Lab. Prove triangles congruent using the
shortcuts we learn from the Triangle Congruence Lab.
We can prove that triangles are congruent by showing that all six pairs of corresponding parts
were congruent. But who wants to go through that every time? Not me!
Triangles are RIGID, giving us a shortcut for proving two triangles are congruent. The question
is how many pairs of corresponding congruent parts does it take before I can conclude that
the triangles are congruent?
2 pairs?
Not enough to guarantee…
3 pairs?
Perhaps. Let’s investigate.
What are the various combos of 3 pairs and which cases force the triangles to be congruent?
6 cases to check: AAA, ASA, SAS, SSS, AAS, SSA
So let’s get our compass and straight-edge so we can complete the Triangle Congruence
Lab! 
Remember to come back and CIRCLE which of the six cases force the triangles to be
congruent once the lab is completed.
Out of the six cases we investigated, which ones actually provide us a shortcut to proving
triangle congruent so that we do not have to find ALL 3 side measures and all 3 angle
measures?
SSA actually ALWAYS works in one specific case…
Given two right triangles, if the hypotenuses are congruent and one pair of legs are
congruent, then by HL (hypotenuse-leg) the triangles are congruent.
AA does not work because the side lengths aren’t specified. You can always stretch or shrink
them.
Examples:
1. BOW   ___________
2. HAT   ___________
H
E
M
O
W
T
I
B
A
Prove the following given the markings in the diagram.
3. Prove NOB  VOB .
O
Statements
Reasons
V
N
B
4. Prove CUE  TUE .
Also given that EU bisects CET .
E
T
U
C
Statements
Reasons
5.
Given: AC UV and B is the midpoint of AU .
Prove: ABC  UBV .
A
Statements
Reasons
Statements
Reasons
V
B
C
U
6.
Given: AC  BD
Prove: ABD  CBD
D
A
B
C
PreAP Geometry Notes
Name __________________________
Section 4-4 & 4-5 EXTENSION CPCTC
Objectives:
Date ________
Use CPCTC to prove parts of congruent triangles are congruent.
CPCTC is an acronym for…
Use CPCTC to complete the following proofs.
1. Given: BP bisects ABC & ABC is isosceles with AB  BC
Prove: P is the midpoint of AC
B
Statements
Reasons
C
P
A
2. Given: B is the midpoint of AD and CE
Prove: CA DE
E
A
B
D
C
Statements
Reasons
PreAP Geometry Notes
Name______________________
Sections 4-6 Isosceles and Equilateral Triangles Date ________
Objectives:
Use properties of isosceles triangles. Use properties of equilateral triangles.
Isosceles Triangle Theorem –
Converse of the Isosceles Triangle Theorem –
Let’s prove the Isosceles Triangle Theorem…
Given: AB  AC
Prove: B  C
Statements:
B
A
Reasons:
C
Example 1:
a. Name the two unmarked congruent
angles.
b. Name two unmarked congruent
segments.
Find the value of x and the measure of A .
Example 2:
A
(3x)˚ (x+44)˚
B
C
What about an equilateral triangle?
If a triangle is equilateral, then it is _______________________________.
What about an equiangular triangle?
If a triangle is equiangular, then it is _______________________________.
Example 3:
a.
Find the value of the variables.
b.
A
J
2t+1
(4y)°
B
(3x+15)°
C
K
4t-8
L