4.1 Triangles and 4.1 Triangles and AnglesAngles

4.1 Triangles and 4.1 Triangles and AnglesAngles

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Standard/Objectives:Objectives:• Classify triangles by their sides and

angles.• Find angle measures in trianglesDEFINITION: A triangle is a figure formed

by three segments joining three non-collinear points.

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Names of triangles

Equilateral—3 congruent sides

Isosceles Triangle—2 congruent sides

Scalene—no congruent sides

Triangles can be classified by the sides or by the angle

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Acute Triangle

mCAB = 41.76 mBCA = 67.97

mABC = 70.26

B

A

C

3 acute angles

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Equiangular triangle• 3 congruent angles. An

equiangular triangle is also acute.

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Right Triangle• 1 right angle • 1 obtuse angle

Obtuse Triangle

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Parts 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 vertext are adjacent sides.

• The third is the side opposite an angle

B

C

A

adjacent

adjacent

Side opposite A

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Right Triangle• Red represents

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

hypotenuseleg

leg

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• 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 Triangles

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Identifying the parts of an isosceles triangle

• 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

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Identifying the parts of an isosceles triangle

• 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

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A

B

C

Using Angle Measures 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.

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Ex. 3 Finding an Angle Measure.

65

x

Exterior Angle theorem: m2 = m A +m 1

1

A 2(2x+10)

x + 65 = (2x + 10)

65 = x +10

55 = x

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Finding angle measures• Corollary to the

triangle sum theorem

• The acute angles of a right triangle are complementary.

• m A + m B = 90

• A

• B

•

2x

x

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Finding angle measuresX + 2x = 903x = 90X = 30

• So m A = 30 and the m B=60

2x

xC

B

A