Chapter 4 Congruent Triangles

In this chapter, you will:classify triangles by their parts,apply the Angle Sum Theorem and the

Exterior Angle Theorem,recognize SSS, SAS, ASA, AAS, and HL

triangle congruence,use properties of isosceles and

equilateral triangles.

4-1: Classifying Triangles

4-1: Classifying Triangles

A triangle is a figure formed by three segments joining three noncollinear points. A triangle can be classified by its sides and by its angles.

sides: AB, BC, ACvertices: A, B, CAngles:<A, <B, <Cm<A+ m<B + m<C = 180°

A

B C

Classification by Sides

Isosceles Triangle

A triangle with at least two sides congruent.

AB = AC<B = <C

A

B C

Scalene Triangle

A triangle with no sides congruent.

Equilateral Triangle

A triangle with three congruent sides.

Classification by angles

Acute Triangle

A triangle with three acute angles.

50°

60° 70°

Right Triangle

A triangle with one right angle.

90°

Obtuse Triangle

A triangle with one obtuse angle.

Equiangular Triangle

A triangle with three congruent angles.

`

Note: An equiangular triangle is also acute and equilateral.

Example 1: Classifying Triangle

Classify triangle ABC

65

58

57

A

B

C

Classify

110

D

F

E

Each of the three points joining the sides of a triangle is a vertex. (The plural of vertex is vertices.) For example, in ∆ ABC, points A, B, and C are vertices.

In a triangle, two sides sharing a common vertex are adjacent sides. In ∆ABC, CA and BA are adjacent sides. The third side, BC, is the side opposite <A.

Side opposite <A

C

B A

Find the measure of each side

9x - 1

4x + 1

5x – 0.5

C

A

B

Assignment

Page 240

Class work: page 240 problems 1-29

Homework: page 241 problems 30-37

Right and Isosceles Triangles: The sides of right triangles and isosceles triangles have special names. In a right triangle, the sides that form the right angle are the legs of the triangle. The side opposite the right angle is the hypotenuse of the triangle.

An isosceles triangle can have three 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 side is the base of the isosceles triangle. Vertex angle

hypotenuse leg

leg

leg leg

base

Theorem 4-2 Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180º.

m<A + m<B + m<C = 180

A

B

C

4-2 Measuring Angles in Triangles

In this section, you will learn:

To apply the Angle Sum Theorem.To apply the Exterior Angle Theorem.

Triangle Angle-Sum Theorem

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

M<A + m<B + m<C =180°.

A

BC

Example 1

Find the measure of <D.

A

D

V41° 74°

Example 2

Find the measure of <A.

A

B60°

Corollary – A statement that can be proved easily using a theorem or a definition.

Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are complementary.

m<A + m<B = 90º

A

C

B

Example 3

Example 4: Finding angle measures

The measure of one acute angle of a right triangle is two times the measure of the other acute angle. Find the measure of each acute angle.

x + 2x = 90

3x = 90

x = 30

M<A = 30º

M<B = 60º

2x

xA

B

Theorem 4-3 Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

m<1 = m<A + m<B

A

B

C

1

Example 4

Find the measure of angle x.

x = 45 + 72x = 117

Example 5

Find the measure of angle x.

x

43

76

Example 6

Example 3: Finding an Angle Measure Find the measure of the exterior angle. First write and solve an

equation to find the value of x.

2x + 10 = x + 65

2x = x + 55

x = 55

So, the measure of the exterior angle is (2 ∙ 55 + 10)º, or 120

x

65

2x + 10

Example 7

Find the value of x, then find the measure of the exterior angle.

2x – 11 = 72 + x

2x = 83 + x

x = 83

2(83) – 11

166 – 11

155

So, the measure of the exterior angle is 155.

x

72 2x-11

Ex. 8

Find the measure of all the angles in the figure.

46°

65°

82°

142°

Example 9

Find the measure of all the angles in the figure.

38°

75°79°

136°

Example 10

Find the measure of all the angles in the figure if AB//CD.

135°

60°

A B

C D

Class work - page 250 (1-29)Homework - page 251 (30-32, 36) and/or handout