Download Triangle Sum

LESSON 3.4
TRIANGLE-SUM THEOREM
During this lesson, you will:
Classify triangles and find the
measures of their angles
Use exterior angles of triangles
Classifying Triangles
Triangles can
be classified
by the lengths
of their sides
and the
measures of
their angles.
VOCABULARY
Isosceles
Scalene
Equilateral
5
12
13
All 3 sides ≡
No sides ≡
At least 2 sides ≡
VOCABULARY
Obtuse
Right
Equiangular
Draw and label an
acute triangle
(Ex. 89,45, 46).
60
60
≡ angles;
each 60º
120
60
Exactly one
90 º angle
Exactly one
angle > 90 º
Classify each triangle below based upon the
lengths of its sides:
EQUILATERAL
SCALENE
ISOSCELES
Classify each triangle based upon
its angle measures:
ACUTE
OBTUSE
RIGHT
EQUIANGULAR
Check for Understanding:
Use your protractor and ruler to draw
and label the following. If not possible,
write not possible.
 isosceles right triangle
 isosceles acute triangle
 isosceles obtuse triangle
Alert!
 An equilateral triangle is always
equiangular.
 An isosceles triangle may be
right, obtuse, or acute.
 A scalene triangle may be right,
obtuse, or acute.
Alert!
 An equiangular triangle is always
equilateral.
 A right triangle may be isosceles or
scalene.
 An obtuse triangle may be isosceles
or scalene.
INVESTIGATION
Write a, b, and c in
the interiors of the
three angles of your
triangle.
Geometry
Carefully cut out
your triangle if it
was not pre-cut for
you.
Tear off the three
angles.
Mrs. McConaughy
Arrange the three
angles so that their
vertices meet at a
fixed point to show
their sum.
10
INVESTIGATIVE RESULTS:
THEOREM
Triangle Sum Conjecture:
The sum of the measures of the
interior angles of any triangle is 180º
▲ Sum = 180.
B
A
mA + m  B + m  C = 180
C
Ex. 1
Z
Find m Z
X
48º
mX + mY + mZ = 180
48 + 67 + mZ = 180
115 + mZ = 180
mZ = 65
67º
Y
Ex. 2
Find the values
of a, b and c.
C
EX. 2
cº 70º
Find c
aº
c + 70 = 90
A
c = 20º
D
Find a. Use Triangle Sum
a + mADC + c = 180
a + 90 + 20 = 180
a + 110 = 180
a = 70º
bº
B
Find b.
Use
Triangle Sum
b = 20º
Use your triangle sum conjecture to calculate
the measure of the third angle in each triangle
below.
Third Angle Conjecture:
If two angles of one triangle are equal in
measure to two angles of another triangle, then
the remaining angles _____________________.
Geometry
Mrs. McConaughy
14
Exterior Angle Theorem
Remote interior
angles
A
B
2
Exterior angle
1
3
x
C
m x = m  1 + m  2
The measure of an
exterior angle in any
triangle equals the
sum of the two
remote interior angles.
Examples:
x
Justify your answer. x = ____
x + y = _____
30
x
y
135
x = ____
x
x
x
90
45
144
x = ____