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
Assignment:
• P. 221-224: 12, 15,
17, 19, 27, 28, 32-39,
48-50, 52-55
• Challenge Problems
Warm-Up
And speaking of the
final frontier, what
do you think is the
shape of space?
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.
3D Rendering
3-D rendering in
digital graphics is
based upon polygons.
Objective 1
You will be able to classify triangles by sides and angles
Vocabulary
As a group, define each
of these types of
triangles without your
book. Draw a picture
for each word and
leave a bit of space
for additions and
revisions.
Classified by Sides
Scalene
Equilateral
Isosceles
Classified by Angles
Acute
Right
Obtuse
Equiangular
Types of Triangles
Objective 2
You will be able to find the measures of the
interior and exterior angles of a triangle
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
Use the following
Investigation to
complete the
Triangle Sum
Theorem.
Investigation 1
1. On a clean sheet of
paper, draw a large
acute triangle.
2. Measure the three
angles of the triangle
as accurately as
possible with your
protractor.
Investigation 1
3. Find the sum of the measures of the
three angles the triangle.
What appears to be the sum of the three
angle measures in every triangle?
Let’s check the sum another way.
Investigation 1
4. Write letters a, b,
and c in the interiors
of the three angles
of the triangle, and
carefully cut out the
triangle.
5. Tear off the three
angles.
Investigation 1
6. Arrange the three
angles so that their
vertices meet at a
point. How does this
arrangement show the
sum of the angle
measures? What is
the sum of the three
angles of any triangle?
Triangle Sum Theorem
The sum of the
measures --?--.
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
40
B
Example 2
Prove the Triangle Sum Theorem.
C
F
1
Given: ABC
a
A
Prove: ma + mb + mc = 180°
c
G
2
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.
Hyperbolic
Elliptic
Non-Euclidean Triangles
Hyperbolic
Elliptic
Big Unanswered Question
So what shape is the universe? Is it basically
flat, or is it curved? How could The
Triangle Sum Theorem help answer this
question?
Big Unanswered Question
So what shape is the
universe? Is it
basically flat, or is
it curved? How
could The Triangle
Sum Theorem
help answer this
question?
Example 3
In the right triangle below, what is the value
of x + y?
y
x
Triangle Sum Theorem Corollary
The acute angles of a right triangle are
complementary.
Investigation 2
Use this Investigation to discover a
relationship between an exterior angle of a
triangle and its two remote interior angles.
Investigation 2
1. On patty paper, draw a
scalene ΔABC. Extend
segment AB through
point B and label a point
D beyond point B. As
shown, put an a in the
interior of <A, a b in the
interior of <B, a c in the
interior of <C, and an x
in exterior angle <CBD.
Investigation 2
2. Copy the two
remote interior
angles <A and <C
onto another patty
paper. Cut the
patty paper into
two pieces, with
an angle on each
piece.
Investigation 2
3. Place the two
remote interior
angles <A and <C
on the exterior
angle to compare
the sum of their
measures against
x, the measure of
the exterior angle.
Investigation 2
How does the sum of the measures of the
two remote interior angles compare with
the measure of the exterior angle?
Exterior Angle Theorem
The measure of
an exterior
angle of a
triangle is equal
to the sum of its
two remote
interior angles.
Example 4
Find m<JKL.
Example 5
Find the
values of a,
b, and c.
75
c
40
120
b
a
Exercise 6: SAT
What is the value of c?
80
c
a
a
b
b
Example 7
Rewrite the Triangle Sum Theorem and its
Corollary in terms of radians.
Example 8
Find the measure of
each angle in
radians.
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
Assignment:
• P. 221-224: 12, 15,
17, 19, 27, 28, 32-39,
48-50, 52-55
• Challenge Problems