TriangleFundamentals

Intro to G.10

Modified by Lisa Palen 1

Triangle

A

B

C

Definition: A triangle is a three-sided polygon.

What’s a polygon?

2

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

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

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

22

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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