Chapter 4 Sec. 1-2:

Classifying Triangles

4.7 Isosceles Triangles

& Equilateral Triangles

Classifying Triangles

• Triangle:

• Polygon: A closed figure in a_____ that is

made up of segments, called _____ that

intersect only at their endpoints, called

______.

• Triangles can be classified by either their

angles or sides:

Terms

A three-sided polygon

plane

sides

vertices

TRIANGLES CLASSIFIED BY ANGLES

Name Definition Example

Acute All angles are

______

Obtuse Only one angle is

______

Right One angle is

______

Equiangular An _______triangle

in which all

angles are

__________

acute

obtuse

right

acute

congruent

60 48

72

30

12030

51

39

60

6060

TRIANGLES CLASSIFIED BY SIDES

Name Definition Example

Scalene All three sides are

_________

Isosceles At LEAST _______

sides are

___________

Equilateral All sides are

_____________

different

two

congruent

congruent

*** REGULAR TRIANGLES are triangles who

are both __________and__________.equiangular equilateral

Parts of an Isosceles Triangle:

Whenever you are solving isosceles triangles, set the two legs

equal to each other or the two base angles equal to each other.

Class Exercises:

1. Triangle RST is an isosceles triangle.

R is the vertex angle, RS = x + 7, ST =

x – 1, and RT = 3x – 5. Find x, RS, ST,

and RT. (Draw a picture!!)R

S T

X+7 3x-5

X-1

X+7=3x-5

12=2x

X=6

RS= x+7=6+7=13

ST= x-1=6-1=5

RT=3x-5=3(6)-5=18-5=13

2. In the figure at the right, Triangle BLM is

isosceles with base ML. Refer to the figure for

the following questions.

a. Identify an acute triangle.

b. Name the hypotenuse

c. Name the vertex angle

d. Name the side opposite C

e. Name the angle opposite MB

f. Name the base angles

g. Name the vertices of the right triangle.

h. Name the legs of the isosceles triangle.

BLM

LM

B

LM

BLM

BLM & BML

L, C, M

BL &BM

3. Fill in the blank with sometimes,

always, or never:

a. Equilateral triangles are ______ isosceles

b. Scalene triangles are _____ isosceles.

c. Right triangles are ______________________ acute.

d. Equiangular triangles are ___________________ acute.

Always

Never

Never

Always

Measuring Angles in Triangles

• Angle Sum Theorem: The sum of the measures of the

angles of a triangle is ___ degrees.

• An _________ is a line or line segment added to a

diagram to help in a proof.

180

auxiliary line

More Theorems

• Third Angle Theorem: if ____ angles of one triangle are _________ to two angles of a second triangle, then the ___________ of the triangles are ________.

• Exterior Angle Theorem: The measure of an _______ angle of a triangle is equal to the _____ of the measures of the two ________________ interior angles.

twoCongruent

third angles congruent

ExteriorSum

Remote

Given: Triangle with angles a, b, and c

Prove: m d = m a + m b

Statements___________________Reasons_________

1. triangle with angles a, b, c 1. given

2. _________________ 2. angle sum thm

3. c and d are linear pairs 3.______________

4. m c + m d = 180 4. ______________

5. m a+m b+m c=m c+m d 5. ______________

6. ______________________ 6. subtraction poe

m a+m b+m c = 180

Def. Of Linear Pairs

Linear Pairs

Substitution

m a+m b=m d

1. If KH is parallel to JI, find the measure of

each angle in the figure.

a. 1

b. 2

c. 3

120-54=66o

(if lines are ll, then alt. Int. angles are congruent)54o

180-54-36=90o

2. For each triangle, find m A. Also tell identify

each triangle by its side and angle measure.

A. B.

C.

Right

2x+21+x+90=180

3x+111=180

3x=69

x=23

m A=2(23)+21=67o

Acute

80+x=3x-22

102=2x

51=x

m A=51o

Obtuse

x+2x+x-20=180

4x-20=180

4x=200

x=50

m A=50-20=30o

3. If AB is perpendicular to BC, find the

measure of each angle in the figure below.

a. 1

b. 2

c. 3

d. 4

e. 5

f. 6

g. 7

h. 8

180-104 = 76

180-36-76 = 68

76

40

180-76-40 = 64

90-64 = 26

180-40 = 140

180-26-140 = 14

Corollary:

• A statement that can be easily proved

using a _____.Theorem

Corollaries:

• The ____ angles of a ____ triangle are

__________.

• There can be at most ___ right or _____ angle

in a triangle.

acute right

complementary

one obtuse

Each angle of an equilateral triangle measures

60 degrees.

Class ExercisesIn isosceles ∆ISO with

base SO, m S = 5x – 18

and m O =2x + 21. Find

the measure of each

angle of the triangle.

Draw a picture!!5x - 18 2x + 21

5x – 18 = 2x + 21

3x = 39

x = 13

m<S = 47

m<O = 47

m<I = 86

Class Exercises

Find the value of x and tell which theorem

you used that we learned this section.

a. b.

62o

xo

2x - 5

10

6

X = 56 by

Isosceles

Triangle

Theorem

10 = 2x – 5

15 = 2x

x = 7.5 by Converse of

Isosceles Triangle Theorem