4.1 Triangles and 4.1 Triangles and AnglesAngles

Equilateral 3 congruent

sidesIsosceles Triangle

2 congruent sides

Scalene 0

congruent sides

Triangles can be classified by the sides or by the angle

Goal 1: Classifying Goal 1: Classifying TrianglesTrianglesA triangle is a figure formed by three segments joining three noncollinear points.

Acute Acute TriangleTriangle

3 acute angles

Classification by Classification by AnglesAngles

B

C

A

Equiangular TriangleEquiangular Triangle

• 3 congruent angles. An equiangular triangle is also acute.

Right Right TriangleTriangle• 1 right angle • 1 obtuse angle

Obtuse Obtuse TriangleTriangle

Parts of a TriangleParts of a Triangle

• Each of the three points joining the sides of a triangle is a vertex. (plural: vertices). A, B and C are vertices.

• Two sides sharing a common vertex are adjacent sides.

• The third is the side opposite an angle

B

C

A

adjacent

adjacent

Side opposite A

When you classify a triangle, you need to be as specific as possible.

Right TriangleRight Triangle• Red represents the

hypotenuse of a right triangle, the side opposite the right angle. The sides that form the right angle are the legs.

hypotenuseleg

leg

• An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is the base.

leg

leg

base

Isosceles Isosceles TrianglesTriangles

Identifying the Parts Identifying the Parts of an Isosceles of an Isosceles TriangleTriangle

• Explain why ∆ABC is an isosceles right triangle.

• In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle.

• Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle.

A B

C

About 7 ft.

5 ft 5 ft

Identifying the parts Identifying the parts of an isosceles of an isosceles triangle triangle (cont.)(cont.)

• Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle?

• Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse.

• Because AC BC, side AB is also the base.

A B

C

About 7 ft.

5 ft 5 ftleg leg

Hypotenuse & Base

A

B

C

Goal 2: Using Angle Goal 2: Using Angle Measures of TrianglesMeasures of Triangles

Smiley faces are interior angles and hearts represent the exterior angles

Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.

Theorems Theorems • Theorem 4.1: Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180°

• Theorem 4.2: Exterior Angle Theorem The measure of an exterior angle of a triangle

is equal to the sum of the measures of the two nonadjacent interior angles

• Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary

Corollary: A statement that can be proved easily using the theorem

Example 3: Finding an Example 3: Finding an Angle MeasureAngle Measure

65(B)

x (A)

Exterior Angle Theorem: m1 = m A +m B

(2x+10) (1)

x + 65 = (2x + 10)65 = x +1055 = x

Finding Angle Finding Angle MeasuresMeasures• Corollary to the

triangle sum theorem

• The acute angles of a right triangle are complementary.

• m A + m B = 90

2x (B)

X (A)

Finding Angle Measures Finding Angle Measures (cont.)(cont.)

X + 2x = 903x = 90X = 30

• So m A = 30 and the m B=60

2x

xC

B

A