Lesson 3-1

Lesson 3-1Triangle

FundamentalsModified by Lisa Palen

Triangle

A

B

C

Definition: A triangle is a three-sided polygon.

What’s a polygon?

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.

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 sidesScaleneIsoscelesEquilateral

Their anglesAcuteRightObtuse Equiangular

Lesson 3-1: Triangle Fundamentals 6

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.

Lesson 3-1: Triangle Fundamentals 7

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.

Lesson 3-1: Triangle Fundamentals 8

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

Classificationof Triangles

with

Flow Chartsand

Venn Diagrams

polygons

Classification by Sides

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral

polygons

Classification by Angles

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangular

Obtuse

obtuse

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?

Parts of Triangles

For example, ∆ABC has

Sides: Angles: A

B

C

Every triangle has three sides and three angles.

ACB

ABC

CABABBCAC

Lesson 3-1: Triangle Fundamentals

14

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

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.

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

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

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 = 180IGO 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

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

Corollary

Each angle in an equiangular triangle measures 60˚.

60

6060

CorollaryThere can be at most one right or obtuse angle in a triangle.

Example

Triangles???

CorollaryAcute 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.

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°

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)

Special Segmentsof Triangles

Introduction

There are four segments associated with triangles:

Medians Altitudes Perpendicular Bisectors Angle Bisectors

Lesson 3-1: Triangle Fundamentals

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.

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

Lesson 3-1: Triangle Fundamentals

30

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

Lesson 3-1: Triangle Fundamentals

31

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