Download Classifying Triangles

Lesson 3-1
Triangle
Fundamentals
Lesson 3-1: Triangle
Fundamentals
1
Naming Triangles
Triangles are named by using its vertices.
For example, we can call the following triangle:
∆ABC
∆ACB
∆BAC
∆BCA
∆CAB
∆CBA
B
C
A
Lesson 3-1: Triangle
Fundamentals
2
Opposite Sides and Angles
Opposite Sides:
A
Side opposite to A : BC
Side opposite to B : AC
Side opposite to C : AB
B
C
Opposite Angles:
Angle opposite to BC : A
Angle opposite to AC : B
Angle opposite to AB : C
Lesson 3-1: Triangle
Fundamentals
3
Classifying Triangles by Sides
Scalene: A triangle in which all 3 sides are different lengths.
A
A
B
C
B
BC = 3.55 cm
C
BC = 5.16 cm
Isosceles: A triangle in which at least 2 sides are equal.
G
Equilateral: A triangle in which all 3 sides are equal.
GH = 3.70 cm
H
Lesson 3-1: Triangle
Fundamentals
HI = 3.70 cm
4
I
Classifying Triangles by Angles
Acute: A triangle in which all 3 angles are less than 90˚.
G
76
57
47
H
Obtuse:
I
A
A triangle in which one and only one
angle is greater than 90˚& less than 180˚
44
28 108 C
B
Lesson 3-1: Triangle
Fundamentals
5
Classifying Triangles by Angles
Right: A triangle in which one and only one angle is 90˚
A
56
B
90
34
C
Equiangular: A triangle in which all 3 angles are the same measure.
B
60
A
60
Lesson 3-1: Triangle
Fundamentals
60
C
6
Classification by Sides
with Flow Charts & Venn Diagrams
polygons
Polygon
triangles
Triangle
scalene
Scalene
Isosceles
isosceles
equilateral
Equilateral
Lesson 3-1: Triangle
Fundamentals
7
Classification by Angles
with Flow Charts & Venn Diagrams
Polygon
polygons
triangles
Triangle
right
acute
Right
Obtuse
Acute
Equiangular
Lesson 3-1: Triangle
Fundamentals
equiangular
obtuse
8
Theorems & Corollaries
Triangle Sum Theorem:
The sum of the interior angles in a
triangle is 180˚.
Third Angle Theorem:
If two angles of one triangle are congruent to two angles of a second
triangle, then the third angles of the triangles are congruent.
Corollary 1: Each angle in an equiangular triangle is 60˚.
Corollary 2: Acute angles in a right triangle are complementary.
Corollary 3: There can be at most one right or obtuse angle in a
triangle.
Lesson 3-1: Triangle
Fundamentals
9
Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles.
Remote Interior Angles
Exterior Angle
mACD  mA  mB
Example: Find the mA.
B
x
A
(3x-22)
D
C
D
B
3x - 22 = x + 80
80
A
3x – x = 80 + 22
C
mA = x = 51°
2x = 102
Lesson 3-1: Triangle
Fundamentals
10
Median - Special Segment of Triangle
Definition: A segment from the vertex of the triangle to the
midpoint of the opposite side.
B
Since there are three vertices, there are three medians.
C
A
F
E
D
In the figure C, E and F are the midpoints of the sides of the triangle.
DC , AF , BE are the medians of the triangle.
Lesson 3-1: Triangle
Fundamentals
11
Altitude - Special Segment of Triangle
Definition: The perpendicular segment from a vertex of the triangle
B
to the segment that contains the opposite side.
C
AF , BE , DC are the altitudes of the triangle.
B
In a right triangle, two of the
altitudes of are the legs of the triangle.
B
I
A
K
A
D
F
E
A
AB, AD, AF  altitudes of right
F
F
D
In an obtuse triangle, two of the altitudes
are outside of the triangle.
BI , DK , AF  altitudes of obtuse
Lesson 3-1: Triangle
Fundamentals
12
D
Perpendicular Bisector – Special
Segment of a triangle
Definition: A line (or ray or segment) that is perpendicular to a
segment at its midpoint.
The perpendicular bisector does not have to start from a vertex!
P
Example:
M
A
E
A
C
D
B
In the scalene ∆CDE, AB
is the perpendicular bisector.
B
O
L
N
In the right ∆MLN, AB is
the perpendicular bisector.
Lesson 3-1: Triangle
Fundamentals
R
In the isosceles
∆POQ, PR is
the perpendicular
bisector.
13
Q