Lesson 3-1: Triangle Fundamentals 1

Lesson 3-1

Triangle Fundamentals

Lesson 3-1: Triangle Fundamentals 2

Naming Triangles

For example, we can call the following triangle:

Triangles are named by using its vertices.

∆ABC ∆BAC

∆CAB ∆CBA∆BCA

∆ACB

A

B

C

Lesson 3-1: Triangle Fundamentals 3

Opposite Sides and Angles

A

B C

Opposite Sides:

Side opposite to A :

Side opposite to B :

Side opposite to C :

Opposite Angles:

Angle opposite to : A

Angle opposite to : B

Angle opposite to : C

BC

AC

AB

BC

AC

AB

Lesson 3-1: Triangle Fundamentals 4

Classifying Triangles by Sides

Equilateral:

Scalene: A triangle in which all 3 sides are different lengths.

Isosceles: A triangle in which at least 2 sides are equal.

A triangle in which all 3 sides are equal.

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

Lesson 3-1: Triangle Fundamentals 5

Classifying Triangles by Angles

Acute:

Obtuse:

A triangle in which all 3 angles are less than 90˚.

A triangle in which one and only one angle is greater than 90˚& less than 180˚

108

44

28 B

C

A

57 47

76

G

H I

Lesson 3-1: Triangle Fundamentals 6

Classifying Triangles by Angles

Right:

Equiangular:

A triangle in which one and only one angle is 90˚

A triangle in which all 3 angles are the same measure.

34

56

90B C

A

60

6060C

B

A

Lesson 3-1: Triangle Fundamentals 7

polygons

Classification by Sides with Flow Charts & Venn Diagrams

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral

Lesson 3-1: Triangle Fundamentals 8

polygons

Classification by Angles with Flow Charts & Venn Diagrams

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangular

Obtuse

obtuse

Lesson 3-1: Triangle Fundamentals 9

Theorems & Corollaries

The sum of the interior angles in a triangle is 180˚.

Triangle Sum Theorem:

Third Angle Theorem:

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Corollary 1: Each angle in an equiangular triangle is 60˚.

Corollary 2: Acute angles in a right triangle are complementary.

Corollary 3: There can be at most one right or obtuse angle in a triangle.

Lesson 3-1: Triangle Fundamentals 10

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

mA = x = 51°

Lesson 3-1: Triangle Fundamentals 11

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

CF

Lesson 3-1: Triangle Fundamentals 12

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 13

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 B

RO Q

P

AB