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Unit 3
Triangles and their properties
Lesson 3.1
Classifying Triangles
Triangle Sum Theorem
Exterior Angle Theorem
Classification of Triangles by
Sides
Name
Equilateral
Isosceles
Scalene
3 congruent
sides
At least 2
congruent sides
No Congruent
Sides
Looks Like
Characteristics
Classification of Triangles by
Angles
Name
Acute
Equiangular
Right
Obtuse
3 acute
angles
3 congruent
angles
1 right
angles
1 obtuse
angle
Looks Like
Characteristics
Example 1
You must classify the triangle as specific as
you possibly can.
 That means you must name
 Classification according to angles
 Classification according to sides
 In that order!
 Example
Obtuse isosceles

More Examples
8.6
600
8.6
600
600
8.6
2.5
1280
260
2.5
260
4.5
Equiangular
Obtuse
Equilateral
Isosceles
And more examples…
Draw a sketch of the following
triangles. Use proper symbol
notation
Obtuse Scalene
Equilateral
Right Equilateral
impossible
Proving the Sum of a triangle’s Angles
What do we know about the two green angles labeled X?
What do we know about the two yellow angles labeled Y?
What do we know about the three angles at the top of the
triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
What do we know about the two yellow angles labeled Y?
What do we know about the three angles at the top of the
triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
What do we know about the three angles at the top of the
triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
X+Z+Y=180 because they form a straight line.
Proving the Sum of a triangle’s Angles
The sum of the
interior angles
must also be
equal to 1800
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
X+Z+Y=180 because they form a straight line.
Triangle Sum Theorem

The sum of the interior angles of a triangle
is 180.
Let’s try some…
m2  48
m1 = 37
m2  57
Proving the Exterior Angle Theorem
ma  mb  c  180 triangle sumtheorem
mb  md  180 Linear Pair
a+b+c=b+d Substitution
a+c=d Subtraction (subtract b from both sides)
Exterior Angle Theorem
Practice
(7x+1)+38=10x+9
7x+39=10x+9
39=3x+9
30=3x
10=x
Lesson 3.2

Inequalities in one triangle
Side/Angle Pairs in a Triangle
B
A
C
Angle A and the side opposite it are a pair
Angle B and the side opposite it are a pair
Angle C and the side opposite it are a pair
The Inequalities in One Triangle
If it is the longest side, then it is opposite
largest angle measure.
 If it is the shortest side, then it is opposite
the smallest angle measure.
 If it is the middle length side, then it is
opposite the middle angle measure.

and their converse, too!
If it is the largest angle measure, then it is
opposite the longest side.
 If it is the smallest angle measure, then it is
opposite the shortest side.
 If it is the middle angle measure, then it is
opposite the middle length side.

Examples: Order the angles from
smallest to largest.
L
M
B
D
K
C
More examples…Order the
angles from smallest to largest
In UVW
VW 12
UW  11
UV  5.8
V
12
5.8
U
11
W
V
U
W
Let’s practice the converse now!
Order the sides from shortest to
longest.
AC
FD
AB
FE and ED
CB
Week’s Schedule







Mon: Lesson 3.3
Tue: Lesson 3.4
Wed: MEAP
Thu: Quiz/Lesson 3.5
Fri: Practice test
Mon: Review Unit 3
Tue: Unit 3 Test
Lesson 3.3

Base Angles Theorem and its Converse
Let’s talk; What do we know
about the following triangles?
Base Angles Theorem and
Converse
Examples: solve for x and/or y
7
If the two angles are equal and the
interior angles of a triangle have a
sum of 180, what is the measure of
the two angles?
75
30
3x = 45
x = 15
y+7 = 45
y = 38
One more example…
solve for x and y
Using the value of x, the
measure of the two angles are
each… 55 degrees
Using the triangle sum theorem
the last angle measure is… 70 degrees
3x-11 = 2x+11
x –11 = 11
x = 22
2y = 70
y = 35
Corollaries (a statement that is
easily proven using the original
theorem)
Prove what the angles in an equilateral
triangle MUST always be.
If all the sides are the same,
equilateral, then all the angles must
be the same, equiangular.
If one of the angles is x, then all
of the angles must also be x.
x + x + x = 180 (triangle sum)
3x = 180 (combine like terms)
x = 60 (DPOE)
Examples
y
5x = 60
x = 12
2x – 3 = 7
2x = 10
x=5
One more!
All sides are equal. Pick any
two and set them equal to each
other. Then solve for x.
12x – 13 = 2x + 17
10x –13 = 17
10x = 30
x=3
Lesson 3.4

Altitudes, Medians, and Perpendicular
Bisectors of Triangles
Putting old terms together…



Perpendicular:
Two lines that intersect at a right angle.
Bisector:
A segment, ray, or line that divides a segment into
two congruent parts.
Perpendicular bisector (of a triangle):
A segment, ray, or line that is perpendicular to a
side of a triangle at the midpoint of the side.
Is segment BD a perpendicular
bisector? Explain!
No, it is
nether perp.
nor a bisector.
No, it is a
bisector but
is not perp.
Yes, it is perp. to
segment AC and
divides it into two
congruent parts.
No, segment
BD is perp. to
segment BC,
but is not its
bisector
New vocabulary terms!
Median of a Triangle: A segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side.
Altitude of a Triangle: The perpendicular
segment from a vertex to the opposite side or
to the line that contains the opposite side
Is segment BD a Median?
Altitude? Explain!
Neither, it is
not a bisector
and it is not
perp.
Median, it is
a bisector of
segment AC
Both, it is a
bisector and is
perp.
D
Altitude, it is
perp. to
segment BC
Special notes about Perp.
Bisectors and Medians:
All perp. bisectors are also medians.
 Some medians are perp. bisectors.
 If it’s not a median, then it is not a perp.
bisector.

Special notes about Perp.
Bisectors and Altitudes:
All perp. bisectors are also altitudes.
 Some altitudes are perp. bisectors.
 If it’s not a altitude, then it is not a perp.
bisector.

Lesson 3.5

Perimeter and Area of Triangles
Review: Identify the altitude of
the following triangles.
Area Formula of a Triangle
1
A  bh
2
b is the base
h is the height
HINT: The base and the height always meet at
a right angle
Find the area of the triangles
1
A  (24)(10)
2
A  12(10)
A  120
1
A  (10.5)(6)
2
A  3(10.5)
A  31.5
Find the area.
8
1
A  (36)(8)
2
A  18(8)
A  144
Formula for the perimeter of a
triangle.
P=a+b+c
a, b, and c are the three sides of the triangle.
HINT: Perimeter is the sum of all three sides
of a trianlge
Find the perimeter of the
triangles
P=26+24+10
P=60
P=6.5+10+10.5
P=27
Find the perimeter
31
P=31+10+36
P=77