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4.1 Apply Triangle Sum Properties
Objectives:
1. To classify triangles by sides and angles
2. To find the measures of the interior and
exterior angles of a triangle
Polygons
A closed plane figure is
a polygon if it is
formed by 3 or more
line segments (sides),
joined endpoint to
endpoint (vertices)
with each side
intersecting exactly
two others.
3-D Rendering
3-D rendering in digital graphics is based
upon polygons.
Vocabulary
In your notebook, define
Classified by Sides
each of these types of
Isosceles
triangles without your Scalene
book. Draw a picture Equilateral
for each word and
leave a bit of space
Classified by Angles
for additions and
Acute
Right
revisions.
Obtuse
Equiangular
Types of Triangles
Triangles in Architecture
Triangles in Architecture
Interior vs. Exterior Angles
All the angles inside the triangle are called
interior angles. If you extend the sides of
the triangle, then the angles that form a
linear pair with the interior angles are
called exterior angles.
Investigation 1
Click on the button to
investigate the
Triangle Sum
Theorem.
Triangle Sum Theorem
The sum of the
measures --?--.
Write and complete this
statement in your notebook
Click the arrow
buttons to
watch the Flash
animation of the
Triangle Sum
Theorem.
Example 1
In ABC below, find the measures of  1
and 2.
1
C
A
60
2
<1 = 80
<2 = 100
40
B
Example 2
Prove the Triangle Sum Theorem.
F
Given: ABC
Prove: ma + mb + mc = 180°
C
1
c
G
2
HINT: Uses parallel lines theorem and
linear angle sum
a
A
TRY THIS in your notebook!
b
B
Non-Euclidean Triangles
Does the Triangle Sum Conjecture hold true
in either hyperbolic or elliptic geometry?
The answer may reveal secrets of the
universe.
C
B
A
Hyperbolic
Elliptic
Big Unanswered Question
So what shape is the universe? Is it basically
flat, or is it curved?
Big Unanswered Question
So what shape is the
universe? Is it
basically flat, or is
it curved?
Example 3
In the right triangle below, what is the value
of x + y?
45
y
x
Triangle Sum Theorem Corollary
The acute angles of a right triangle are
complementary.
Exterior Angle Theorem:
The measure of an
exterior angle of a
triangle is equal to
the sum of its two
remote interior
angles.
remote interior
Example 4
Find m<JKL.
m<JKL = 35
Example 5
Find the
values of a,
b, and c.
75
c
40
<a = 65
<b = 60
<c = 55
120
b
a
Exercise 6: SAT
What is the value of c?
80
c
a
a
160
b
b
Example 7
Rewrite the Triangle Sum Theorem and its
Corollary in terms of radians.
Do this in your notebook
Example 8
Find the measure of
each angle in radians.
<A & <C = 2π
7
<B = 3π
7