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Triangle Relationships
Chapter 4
Objectives:
 Classifying triangles and finding their
angle measures.
 Using the Distance Formula, the
Pythagorean Theorem, and its converse.
 Showing relationships between a triangle’s
sides and angles.
Sections
 4.1 Classifying Triangles
 4.2 Angle Measures of Triangles
 4.3 Isosceles and Equilateral Triangles
 4.4 The Pythagorean Theorem and the
Distance Formula
 4.5 The Converse of the Pythagorean
Theorem
 4.6 Medians of a Triangle
 4.7 Triangle Inequalities
Classifying Triangles
Section 4.1
Objectives
 Identify and classify triangles by angles
 Identify and classify triangles by sides
Key Vocabulary
 Acute Triangle
 Equiangular Triangle
 Obtuse Triangle
 Right Triangle
 Equilateral Triangle
 Isosceles Triangle
 Scalene Triangle
 Vertex
Definition
 Triangle: a figure formed by three segments
joining three noncollinear points.
B
A
C
 Two methods of classifying or naming triangles:
 Angles
 Sides
 Triangle notation: ∆
 Name triangle using three letters, therefore the
above triangle is ∆ABC.
Parts of a Triangle
 A triangle is a 3-sided polygon
 The sides of ∆ABC are





A
AB, BC, and AC
A vertex of a triangle is a point that adjacent
joins two sides of the triangle.
The vertices of ∆ABC are
B
A, B, and C
Two sides sharing a common vertex are
adjacent sides
The third side is called the opposite side
All sides can be adjacent or opposite (it just
depends which vertex is being used)
adjacent
Side
opposite
A
C
Example 1:
Name the side that is opposite the angle.
a. A
b. B
SOLUTION
a. BC is the side that is opposite A.
b. AC is the side that is opposite B.
c. AB is the side that is opposite C.
c. C
Classifying Triangles by Angles
 Four Classifications




Acute
Obtuse
Right
Equiangular
 All triangles have at least two acute
angles, the third angle is used to classify
the triangle.
Classifying Triangles by Angles
One way to classify triangles is
by their angles…
Acute
Obtuse
all 3 angles are acute 1 angle is obtuse
(measure < 90°)
(measure > 90°)
)
)
(
Right
1 angle is right
(measure = 90°)
Equiangular
All 3 congruent acute angles
(measure < 90˚ and ≅)
Classifying Triangles by Angles
Definition: ACUTE Triangle
a triangle in which all angles
are acute.
E
30
D
70
80
F
Classifying Triangles by Angles
40
E
30
110 
F
Definition: OBTUSE Triangle
a triangle in which one of the
angles is an obtuse angle.
D
Classifying Triangles by Angles
A
Hypotenuse
B
Definition: RIGHT Triangle
a triangle in which one of the
angles is a right angle.
Leg
Leg
C
Classifying Triangles by Angles
D
Definition:
EQUIANGULAR Triangle
a triangle in which all
angles are congruent.
60
60 60
F
E
**EQUIANGULAR applies to any figure
in which all angles are congruent**
Classifying Triangles by Angles
 Each of the classifications (acute, obtuse,
right, equiangular) is a distinct group and
should not be combined.
 A common mistake is to place triangles
into more than one of the angle
classifications.
 Example: a right triangle cannot be
classified as an acute triangle.
Example 2:
The triangular truss below is modeled for steel
construction. Classify
JMN, JKO, and OLN as acute, equiangular, obtuse,
or right.
Example 2:
Answer:
JMN has one angle with measure greater than 90, so it
is an obtuse triangle.
JKO has one angle with measure equal to 90, so it is a
right triangle.
OLN is an acute triangle with all angles congruent, so it
is an equiangular triangle.
Classifying Triangles by Sides
 Triangles can also be classified according to the
number of congruent sides they have.
 Three classifications:
 Equilateral
 Isosceles
 Scalene
 To indicate that sides of a triangle are
congruent, an equal number of hash marks is
drawn on the corresponding sides.
Classifying Triangles by Sides
Another way to classify triangles
is by their sides…
Equilateral
3 congruent sides
Isosceles
Scalene
2 congruent sides no congruent sides
Classifying Triangles by Sides
D
Definition:
EQUILATERAL
a triangle in which all
sides are congruent.
E
F
**EQUILATERAL applies to any figure in
which all sides are congruent**
Classifying Triangles by Sides
Vertex Angle A
Definition: ISOSCELES
a triangle in which at
least 2 sides are
congruent.
Leg
B
Leg
Base
Base Angles
C
Classifying Triangles by Sides
M
Definition: SCALENE
a triangle in which no
sides are congruent.
O
N
Example 3:
Classify the triangle by its sides.
a.
b.
c.
SOLUTION
a. Because this triangle has 3 congruent sides, it is
equilateral.
b. Because this triangle has no congruent sides, it is
scalene.
c. Because this triangle has 2 congruent sides, it is
isosceles.
Your Turn:
Classify the triangle by its sides.
1.
ANSWER
isosceles
2.
ANSWER
equilateral
ANSWER
scalene
3.
Example 4:
Name the side that is opposite the angle.
a. A
b. B
c. C
SOLUTION
a. BC is the side that is opposite A.
b. AC is the side that is opposite B.
c. AB is the side that is opposite C.
Example 5:
Identify the isosceles triangles in the figure if
Isosceles triangles have at least two sides congruent.
Answer: UTX and UVX are isosceles.
Example 6:
Identify the scalene triangles in the figure if
Scalene triangles have no congruent sides.
Answer: VYX, ZTX, VZU, YTU, VWX,
ZUX, and YXU are scalene.
Your Turn:
Identify the indicated triangles in the figure.
a. isosceles triangles
Answer: ADE, ABE
b. scalene triangles
Answer: ABC, EBC, DEB, DCE, ADC, ABD
Example 7:
ALGEBRA Find d and the measure of each side of
equilateral triangle KLM if
and
Since KLM is equilateral,
each side has the same
length. So
5=d
Example 7:
Next, substitute to find the length of each side.
KL = 7
LM = 7
KM = 7
Answer: For KLM,
and the measure of
each side is 7.
Your Turn:
ALGEBRA Find x and the measure of each side of
equilateral triangle
if
and
Answer:
Review: Classifications of
Triangles by Angles
polygons
Polygon
triangles
Triangle – 3 sides
right
acute
Right
One 90˚ ∠
Acute
Obtuse
One ∠ > 90˚ All ∠s < 90˚
Equiangular
All ∠s ≅
equiangular
obtuse
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Review: Classifications of
Triangles by Sides
polygons
Polygon
Triangle – 3 sides
Scalene
No sides ≅
Isosceles
2 sides ≅
Equilateral
3 sides ≅
triangles
scalene
isosceles
equilateral
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Joke Time
 What did the pony say when he had a cold?
 I’m just a little horse!
 What is Beethoven doing in his grave?
 De-composing
 What do you call an arrogant household bug?
 A cocky roach.
Assignment
 Sec. 4.1 Pg. 175-178: #1 – 29 odd, 30 –
36 all, 37 – 65 odd