Transcript Intro to G.10 Triangle Fundamentals

Intro to
G.10
Triangle
Fundamentals
Modified by Lisa Palen
1
2
Triangle
Definition: A triangle is a three-sided polygon.
B
What’s a polygon?
C
A
Polygons
3
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
4
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.
5
Triangles can be classified
by:
Their sides
Scalene
Isosceles
Equilateral
Their angles
Acute
Right
Obtuse
Equiangular
6
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
HI = 3.70 cm
I
7
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
90
34
C
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Classifying Triangles by Angles
Acute:
G
76
A triangle in which all three angles are....
acute.
57
H
Equiangular:
A triangle in which all three angles are...
congruent.
47
I
Classification
of Triangles
with
Flow Charts
and
Venn Diagrams
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10
Classification by Sides
polygons
Polygon
triangles
Triangle
scalene
Scalene
Isosceles
isosceles
equilateral
Equilateral
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Classification by Angles
Polygon
polygons
triangles
Triangle
right
acute
Right
Obtuse
Acute
equiangular
obtuse
Equiangular
12
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?
13
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
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Opposite Sides and Angles
Opposite Sides:
Side opposite of BAC
:
Side opposite of ABC
:
Side opposite of ACB
:
Opposite Angles:
A
BC
AC
AB
Angle opposite of BC : BAC
Angle opposite of AC : ABC
Angle opposite of AB : ACB
B
C
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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
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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
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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
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19
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
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Third Angle Corollary
If two angles in one triangle are
congruent to two angles in
another triangle, then the third
angles are congruent.
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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
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Corollary
Each angle in an equiangular
triangle measures 60˚.
60
60
60
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Corollary
There can be at most one right
or obtuse angle in a triangle.
Example
Triangles???
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Corollary
Acute angles in a right triangle
are complementary.
Example
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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°
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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
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Introduction
There are four segments associated with
triangles:
 Medians
 Altitudes
 Perpendicular Bisectors
 Angle Bisectors
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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.
C
B
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 medians of the triangle.
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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.
BI , DK , AF  altitudes of obtuse
D
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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
In the right ∆MLN, AB is
the perpendicular bisector.
R
In the isosceles
∆POQ, PR is
the perpendicular
bisector.
Q