Triangles

1

The Basics

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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

3

Opposite Sides and Angles

A

B C

Opposite Sides:

Side opposite to angle A

Side opposite to angle B

Side opposite to angle C

Opposite Angles:

Angle opposite to : angle A

Angle opposite to : angle B

Angle opposite to : angle C

BC

AC

AB

BC

AC

AB

Classifying Triangles by Sides

Equilateral:

4

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.

3.2

cm

3.15 cm

C 3.55 cm

A

B C

3.47

cm

3.47 cm

5.16 cmBC

A

G

H I

3.7 cm

3.7 cm

3.7 cm

Classifying Triangles by Angles

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

5

Acute:

Obtuse:

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

Classifying Triangles by Angles

6

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°

90°B C

A

60°

60°60°C

B

A

7

polygons

Classification by Sides with Flow Charts & Venn Diagrams

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral

8

polygons

Classification by Angles with Flow Charts & Venn Diagrams

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangular

Obtuse

obtuse

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.

10

Exterior Angle TheoremThe 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)°x°80°

B

A DC

Solve for x.

3x - 22 = x + 80

3x – x = 80 + 22

2x = 102m<A = x = 51°

Classifying TrianglesThere are 3 ways

to classify a triangle by its sides.ScaleneIsoscelesEquilateral

There are 4 ways to classify a triangle by its angles.AcuteRightObtuseEquiangular

** Equiangular Triangles are always Equilateral.