Download Lesson 3-1

Lesson
Triangle 3-1
Fundamentals
Modified by Lisa Palen
Triangle
Definition: A triangle is a three-sided polygon.
B
What’s a polygon?
C
A
Polygons
Definition: A closed figure formed by a finite number of coplanar
segments so that each segment intersects exactly two
others, but only at their endpoints.
These figures are not polygons
These figures are polygons
Definition of a Polygon
A polygon is a closed figure in
a plane formed by a finite
number of segments that
intersect only at their
endpoints.
From Lesson 3-4
Triangles can be classified
by:
Their sides
Scalene
Isosceles
Equilateral
Their angles
Acute
Right
Obtuse
Equiangular
Classifying Triangles by Sides
Scalene: A triangle in which no sides are congruent.
A
A
B
BC = 3.55 cm
C
B
C
BC = 5.16 cm
Isosceles: A triangle in which at least 2 sides are congruent.
Equilateral:
A triangle in which all 3 sides are congruent.
G
GH = 3.70 cm
H
Lesson 3-1: Triangle Fundamentals
6
HI = 3.70 cm
I
Classifying Triangles by Angles
Obtuse:
A
44
A triangle in which one angle is....
obtuse.
28 108 C
B
Right:
A triangle in which one angle is...
A
56
right.
B
Lesson 3-1: Triangle Fundamentals
90
34
7
C
Classifying Triangles by Angles
Acute:
G
76
A triangle in which all three angles are....
acute.
57
47
H
Equiangular:
I
A triangle in which all three angles are...
congruent.
Lesson 3-1: Triangle Fundamentals
8
Classification
of Triangles
with
Flow Charts
and
Venn Diagrams
Classification by Sides
polygons
Polygon
triangles
Triangle
scalene
Scalene
Isosceles
isosceles
equilateral
Equilateral
Classification by Angles
Polygon
polygons
triangles
Triangle
right
acute
Right
Obtuse
Acute
equiangular
obtuse
Equiangular
Naming Triangles
B
We name a triangle using
its vertices.
For example, we can call this
triangle:
∆ABC
∆ACB
∆BAC
∆BCA
∆CAB
∆CBA
C
A
Review: What is ABC?
Parts of Triangles
B
Every triangle has three
sides and three angles.
For example, ∆ABC has
Sides:
AB
BC
AC
Angles:
 CAB
 ABC
 ACB
C
A
14
Opposite Sides and Angles
Opposite Sides:
Side opposite of BAC
:
Side opposite of ABC
:
Side opposite of ACB
:
Opposite Angles:
A
BC
AC
AB
B
Angle opposite of BC : BAC
Angle opposite of AC : ABC
Angle opposite of AB : ACB
Lesson 3-1: Triangle Fundamentals
C
Interior Angle of a Triangle
An interior angle of a triangle (or any polygon)
is an angle inside the triangle (or polygon),
formed by two adjacent sides.
For example, ∆ABC has interior
angles:
B
 ABC,  BAC,  BCA
C
A
Exterior Angle
An exterior angle of a triangle (or any polygon)
is an angle that forms a linear pair with an interior
angle. They are the angles outside the polygon
formed by extending a side of the triangle (or
polygon) into a ray.
Interior Angles
For example, ∆ABC has
exterior angle ACD,
because ACD forms a
linear pair with ACB.
Exterior Angle
A
D
B
C
Interior and Exterior Angles
The remote interior angles of a triangle (or any
polygon) are the two interior angles that are “far
away from” a given exterior angle. They are the
angles that do not form a linear pair with a given
exterior angle.
For example, ∆ABCRemote
has Interior
exterior angle:
Angles
Exterior Angle
A
ACD and
remote interior angles
A and B
D
B
C
Triangle
Theorems
Triangle Sum Theorem
The sum of the measures of the
interior angles in a triangle is 180˚.
m<A + m<B + m<C = 180
IGO GeoGebra Applet
Third Angle Corollary
If two angles in one triangle are
congruent to two angles in
another triangle, then the third
angles are congruent.
Third Angle Corollary Proof
Given: The diagram
B
A
Prove: C  F E
C
D
statements
1. A  D, B  E
2. mA = mD, mB = mE
3. mA + mB + m C = 180º
mD + mE + m F = 180º
4. m C = 180º – m A – mB
m F = 180º – m D – mE
5. m C = 180º – m D – mE
6. mC = mF
7. C  F
QED
reasons
1. Given
2. Definition: congruence
3. Triangle Sum Theorem
4.
Subtraction Property of Equality
5.
6.
7.
Property: Substitution
Property: Substitution
Definition: congruence
F
Corollary
Each angle in an equiangular
triangle measures 60˚.
60
60
60
Corollary
There can be at most one right
or obtuse angle in a triangle.
Example
Triangles???
Corollary
Acute angles in a right triangle
are complementary.
Example
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
3x - 22 = x + 80
80
x
A
(3x-22)
D
C
3x – x = 80 + 22
2x = 102
x = 51
A
D
B
C
mA = x = 51°
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the remote interior angles.
GeoGebra Applet (Theorem 1)
Special Segments
of
Triangles
Introduction
There are four segments associated with
triangles:
 Medians
 Altitudes
 Perpendicular Bisectors
 Angle Bisectors
29
Median - Special Segment of Triangle
Definition: A segment from the vertex of the triangle to the
midpoint of the opposite side.
Since there are three vertices, there are three medians.
B
C
F
D
E
A
In the figure C, E and F are the midpoints of the sides of the triangle.
DC , AF , BE are the mediansLesson
of 3-1:
theTriangle
triangle
.
Fundamentals
30
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.
In a right triangle, two of the
altitudes are the legs of the triangle.
B
A
K
E
A
A
D
F
F
AB, AD, AF  altitudes of right
F
I
B
D
In an obtuse triangle, two of the altitudes
are outside of the triangle.
Lesson 3-1: of
Triangle
Fundamentals
BI , DK , AF  altitudes
obtuse
D
31
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:
E
M
A
A
C
D
B
In the scalene ∆CDE, AB
is the perpendicular bisector.
L
B
O
N
R
In the isosceles
∆POQ, PR is
In the right ∆MLN, AB is the perpendicular
the perpendicular bisector.
bisector.
Lesson 3-1: Triangle Fundamentals
Q