Geometry - 4.1/4.2

Classifying Triangles and the Angles of

Triangles

Classifying TrianglesYou can classify triangles in two ways: by their sides, or by their angles.

Classification by Sides

Classification by Angles

Equilateral Isosceles Scalene

3 Congruent Sides At least 2 Congruent Sides No Congruent Sides

Acute Equiangular ObtuseRight

3 Acute Angles 3 Congruent Angles 1 Right Angle 1 Obtuse Angle

DefinitionsEach point that joins the sides of a triangle is called a vertex.

Adjacent sides of a triangle are sides that share a common vertex.

If you choose any one vertex, and note the two adjacent sides, the side that is not adjacent is known as the opposite side.

Assume A is the vertex we are referencing

DefinitionsIn a right triangle, the adjacent sides to the vertex where the right angle is located, are called the legs of a right triangle.

The opposite side from the right angle is called the hypotenuse of the right triangle.

In an isosceles triangle, where only two sides are congruent, the congruent sides are called the legs of the isosceles triangle.

The non-congruent side of the isosceles triangle is called the base.

Triangle Sum Theorem Thm.– The sum of the measures of the interior angles of a triangle is 180 degrees.

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

A corollary to a theorem is a statement that can be easily proven using the given theorem.

Corollary– The acute angles of a right triangle are complementary.

m<B + m<C = 90

Concept 1

Proof of the Triangle Sum Thm

• 1) Draw BD parallel to AC• 2) m<4 + m<2 + m<5 = 180• 3) • 4) m<1 = m<4, m<3 = m<5• 5) m<1 + m<2 + m<3 = 180

• 1) Parallel Postulate• 2) Def of straight angle• 3) Alt. Int. <‘s Thm.• 4) Def of congruent <‘s• 5) Substitution Property

1 4, 3 5

Given: Triangle ABC

Prove: m<1 + m<2 + m<3 = 180

Use the Triangle Angle-Sum Theorem

SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.

Answer: Therefore, m1 = 63, m2 = 63, and m3 = 38.

Check The sums of the measures of the angles in each triangle should be 180.

m1 + 43 + 74 = 63 + 43 + 74 or 180m2 + m3 + 79 = 63 + 38 + 79 or 180

A. 95

B. 75

C. 57

D. 85

Find the measure of 3.

Exterior Angle Thm

72 2 11

83

x x

x

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

m<1 = m<A + m<B

Find the value of the exterior angle shown.

Use the Exterior Angle Theorem

GARDENING Find the measure of FLW in the fenced flower garden shown.

mLOW + mOWL= mFLW Exterior Angle Theorem

x + 32 = 2x – 48 Substitution

32 = x – 48 Subtract x from each side.

80 = x Add 48 to each side.Answer: So, mFLW = 2(80) – 48 or 112.

A. 30

B. 40

C. 50

D. 130

The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD.

Using Corollaries to Find Angles

11 , 2

4m x m x

1 5 36090 90 72

4 4 5x x x x x

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.

The measure of one acute angle of a right triangle is one-fourth the measure of the other acute angle. Find the measure of each acute angle.

m<1 = x, m<2 = 2x x + 2x = 90 3x = 90 x = 30

m<1 = 30, m<2 = 60

m<1 = 72, m<2 = 18

Step 1 Find d.

ALGEBRA Find the measure of the sides of isosceles triangle KLM with base KL.

__

KM = ML

4d – 13= 12 – d

5d – 13= 12

5d = 25

d = 5

Answer: KM = ML = 7, KL = 11

Step 2

Substitute to find the length of each side.

KM = 4d – 13

= 4(5) – 13 or 7

ML = KM

= 7

KL = d + 6

= 5 + 6 or 11

ALGEBRA Find x and the measure of each side of equilateral triangle ABC if AB = 6x – 8, BC = 7 + x, and AC = 13 – x.

A. x = 10; all sides are 3.

B. x = 6; all sides are 13.

C. x = 3; all sides are 10.

D. x = 3; all sides are 16.

Find the measure of each numbered angle.

m1 = 104, m2 = 76, m 3 = 42, m4 = 48, m5 = 49

A. 50

B. 45

C. 85

D. 130

Find m3.

Practice Problems

•Pg. 239 #15-37(odds)

•Pg.248 #1-31(odds),34-37

•HW Check Next Class