Triangle Fundamentals Intro to G.10 Modified by Lisa Palen 1.
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Transcript of Triangle Fundamentals Intro to G.10 Modified by Lisa Palen 1.
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TriangleFundamentals
Intro to G.10
Modified by Lisa Palen 1
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Triangle
A
B
C
Definition: A triangle is a three-sided polygon.
What’s a polygon?
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These figures are not polygons These figures are 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.
Polygons3
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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.
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Triangles can be classified by:
Their sidesScaleneIsoscelesEquilateral
Their anglesAcuteRightObtuse Equiangular
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Classifying Triangles by Sides
Equilateral:
Scalene: A triangle in which no sides are congruent.
Isosceles:
AB
= 3
.02
cm
AC
= 3.15 cm
BC = 3.55 cm
A
B CAB =
3.47
cmAC = 3.47 cm
BC = 5.16 cmBC
A
HI = 3.70 cm
G
H I
GH = 3.70 cm
GI = 3.70 cm
A triangle in which at least 2 sides are congruent.
A triangle in which all 3 sides are congruent.
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Classifying Triangles by Angles
Obtuse:
Right:
A triangle in which one angle is....
A triangle in which one angle is...
108
44
28 B
C
A
34
56
90B C
A
obtuse.
right.
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Classifying Triangles by Angles
Acute:
Equiangular:
A triangle in which all three angles are....
A triangle in which all three angles are...
acute.
congruent.
57 47
76
G
H I
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Classificationof Triangles
with
Flow Chartsand
Venn Diagrams9
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polygons
Classification by Sides
triangles
Scalene
Equilateral
Isosceles
Triangle
Polygon
scalene
isosceles
equilateral
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polygons
Classification by Angles
triangles
Right
Equiangular
Acute
Triangle
Polygon
right
acute
equiangular
Obtuse
obtuse
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Naming Triangles
For example, we can call this triangle: A
B
C
We name a triangle using its vertices.
∆ABC
∆BAC
∆CAB ∆CBA
∆BCA
∆ACBReview: What is ABC?
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Parts of Triangles
For example, ∆ABC has
Sides: Angles: A
B
C
Every triangle has three sides and three angles.
ACB
ABC
CABABBCAC
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Opposite Sides and Angles
A
B C
Opposite Sides:
Side opposite of BAC :
Side opposite of ABC :
Side opposite of ACB :
Opposite Angles:
Angle opposite of : BAC
Angle opposite of : ABC
Angle opposite of : ACB
BC
AC
AB
BC
AC
AB
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Interior Angle of a Triangle
For example, ∆ABC has interior angles:
ABC, BAC, BCA
A
B
C
An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides.
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Interior Angles
Exterior Angle
For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB.
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.
A
BC
D
Exterior Angle
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Interior and Exterior Angles
For example, ∆ABC has exterior angle:
ACD and
remote interior angles A and B
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.
A
BC
D
Exterior AngleRemote Interior Angles
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Triangle Theorems
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Triangle Sum Theorem
The sum of the measures of the interior angles in a triangle is 180˚.
m<A + m<B + m<C = 180IGO 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
The diagramGiven:
statements reasons
E
DA
B
CF
Prove: C F
1. A D, B E2. mA = mD, mB = mE3. mA + mB + m C = 180º mD + mE + m F = 180º4. m C = 180º – m A – mB m F = 180º – m D – mE5. m C = 180º – m D – mE6. mC = mF 7. C F
1. Given2. Definition: congruence3. Triangle Sum Theorem
4. Subtraction Property of Equality
5. Property: Substitution6. Property: Substitution7. Definition: congruenceQED
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Corollary
Each angle in an equiangular triangle measures 60˚.
60
6060
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CorollaryThere can be at most one right or obtuse angle in a triangle.
Example
Triangles???
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CorollaryAcute 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.
Exterior AngleRemote Interior Angles A
BC
D
m ACD m A m B
Example:
(3x-22)x80
B
A DC
Find the mA.
3x - 22 = x + 80
3x – x = 80 + 22
2x = 102
x = 51
mA = x = 51°
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Exterior Angle TheoremThe 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)
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Special Segmentsof 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.
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
B
A DE
C F
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Altitude - Special Segment of Triangle
Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.
In a right triangle, two of the altitudes are the legs of the triangle.
B
A DE
C
FB
A D
F
In an obtuse triangle, two of the altitudes are outside of the triangle.
, , .AF BE DC are the altitudes of the triangle
, ,AB AD AF altitudes of right B
A D
F
I
K , ,BI DK AF altitudes of obtuse
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Perpendicular Bisector – Special Segment of a triangle
AB PR
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!
Example:
C D
In the scalene ∆CDE, is the perpendicular bisector.
In the right ∆MLN, is the perpendicular bisector.
In the isosceles ∆POQ, is the perpendicular bisector.
EA
B
M
L N
A BR
O Q
P
AB
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