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Geometry Lesson 4.1
Triangles and
Properties of Their Angles
Warm-Up: Properties of Angles
Vertex: B
Sides: BA, BC
Vertex: Y
60° = Acute
Sides: YX, YZ
180° = Straight
Vertex: L
Vertex: P
Sides: LM, LN
Sides: PR, PQ
90° = Right
125° = Obtuse
1. Definition of a Triangle


A triangle is a figure with three line
segments joining three non-collinear points
Called “triangle ABC”, or in symbols ABC
B
Each point is called a vertex
C Side BC is an
A
opposite side to A
Two sides that share a common
vertex are adjacent sides
(AB and AC are adjacent sides)
Example 1: Defining a Triangle
What is the name of the triangle?
DEF or EFD or FDE
 What are the sides adjacent to E?
ED and EF
 What is the side opposite to E? DF

D
E
F
2. Classifying Triangles

A triangle is classified by its sides and
angles
By Sides
leg
Equilateral
Three
congruent
sides
leg
base
Isosceles
At least two
congruent
sides
Scalene
No
congruent
sides)
Classifying Triangles, cont.
By Angles
leg
Acute
Three
acute angles
Equiangular
Three
congruent s
(is also acute)
Obtuse
One
obtuse angle
leg
Right
One
right angle
Example 2: Classifying Triangles

When classifying a triangle, be as specific
E
as possible
(a)
B
(b)
71°
F
65°
44°
C
A
Sides: None are 
Angles: All are acute
none are 
“ABC is acute scalene”
D
Sides: Two are 
Angles: One is obtuse
“DEF is obtuse isosceles”
Practice 2: Classifying Triangles

Classify each triangle
I
(a)
(b)
88°
46°
46°
J
H
Sides: Two are congruent
Angles: All are acute
Classification:
HIJ is acute isosceles
L
4
K
5
M
3
Sides: None are 
Angles: One is right
Classification:
KLM is right scalene
3. Interior and Exterior Angles
The angles inside the triangle are
called interior angles
 The angles outside the triangle
adjacent to each interior angle are
called exterior angles

Interior Angles
Exterior Angles
B
A
B
C
A
C
4. Triangle Sum Theorem

The sum of the measures of interior
angles is 180°
B
A
180°
VERY
important!
C
mA + mB + mC = 180°
5. Corollary to Triangle Sum Theorem

The acute angles of a right triangle are
complementary
B
Corollary to a
theorem: A statement
that can be proved
easily using the
theorem
Acute
angles
C
A
mA + mC = 90°
6. Exterior Angles Theorem

The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two non-adjacent
interior angles
B
m1 = mA + mB
A
1
C
Example 3: Finding an Angle Measure

Find the value of x
B
72°
A
x°
Exterior angles theorem:
2x + 10 = x + 72
2x – x = 72 – 10
x = 62
(2x+10)°
C
Practice 3: Find Angle Measures

Solve for each variable
(a)
If rt. , then acute s
comp.
3x + 11 + 4x – 5 = 90
7x + 6 = 90
7x = 84 x = 12
(b)
If ext. , then m =
sum of non-adj ms
2y = 90 + 50
2y = 140
y = 70
Ex: 1What is the missing angle?
70º
70º
+ x
180º
x
70º
70º
180 – 140 = 40˚
EX: 2 What is the missing
angle?
30º
90º
x
30º
+
x
90º 180º
180 – 120 = 60˚
Find all the angle measures
35x
45x
180 = 35x + 45x + 10x
180 = 90x
2=x
10x
90°, 70°, 20°
Find the missing angles.
SOLUTION:
2x + x = 90
3x = 90
x = 30˚
2x = 60˚
Closure

Convert the words into a picture, then solve
for the angle measures
“The interior angle measures of a triangle
are x°, 2x°, and 3x°. Find the value of x”
Solution:
x + 2x + 3x = 180
6x = 180
x = 30
x°
30°
90°
3x°
60°
2x°
Assignment
4.1 (pg.198-201)
#10-26, 32-38, 52-68 ALL ARE EVENS