4.1 Triangles & Angles4.1 Triangles & Angles

August 15, 2013August 15, 2013

4.1 Classifying Triangles

Triangle – A figure formed when three noncollinear points are connected by segments.

E

DF

Angle

SideVertex

The sides are DE, EF, and DF.The vertices are D, E, and F.The angles are D, E, F.

Triangles Classified by AnglesAcute Obtuse Right

60º

50º

70º

All acute anglesOne obtuse angle

One right angle

120º

43º

17º

30°

60º

Triangles Classified by Sides

Scalene Isosceles Equilateral

no sidescongruent at least two

sides congruent

all sidescongruent

Classify each triangle by its angles and by its sides.

60°

60° 60°A B

C

45°

45°

E

F G

EFG is a right

isosceles triangle.

ABC is an acute

equilateral triangle

Fill in the tableAcute Obtuse Right

Scalene

Isosceles

Equilateral

Try These:1. ABC has angles that

measure 110, 50, and 20. Classify the triangle by its angles.

2. RST has sides that measure 3 feet, 4 feet, and 5 feet. Classify the triangle by its sides.

Adjacent Sides- share a vertex ex. The sides DE & EF are adjacent to E.

E

D F

Opposite Side- opposite the vertex ex. DF is opposite E.

Parts of Isosceles Triangles The angle formed by the congruent sides is called the vertex angle.

leg leg The congruent sides are called legs.

The side opposite the vertex is the base.

base anglebase angle

The two angles formed by the base and one of the congruent sides are called base angles.

Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent.

If , thenACAB CB

Converse of Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.

If , thenCB ACAB

EXAMPLE 1 Apply the Base Angles Theorem

P

R

Q

(30)°

Find the measures of the angles.SOLUTION

Since a triangle has 180°, 180 – 30 = 150° for the other two angles.

Since the opposite sides are congruent, angles Q and P must be congruent.

150/2 = 75° each.

EXAMPLE 2 Apply the Base Angles Theorem

P

R

Q(48)°

Find the measures of the angles.

EXAMPLE 3 Apply the Base Angles Theorem

P

R

Q(62)°

Find the measures of the angles.

EXAMPLE 4 Apply the Base Angles Theorem

Find the value of x. Then find the measure of each angle.

P

RQ(20x-4)°

(12x+20)° SOLUTION

Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4

20 = 8x – 4

24 = 8x

3 = x

Plugging back in,

And since there must be 180 degrees in the triangle,

564)3(20

5620)3(12

Rm

Pm

685656180Qm

EXAMPLE 5 Apply the Base Angles Theorem

Find the value of x. Then find the measure of each angle.

P

R

Q(11x+8)° (5x+50)°

EXAMPLE 6 Apply the Base Angles Theorem

Find the value of x. Then find the length of the labeled sides.

P

R

Q(80)° (80)°

SOLUTION

Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40

4x = 40

x = 10

7x 3x+40

Plugging back in,

QR = 7(10)= 70PR = 3(10) + 40 = 70

EXAMPLE 7 Apply the Base Angles Theorem

Find the value of x. Then find the length of the labeled sides.

P

RQ

(50)°

(50)°

10x – 2

5x+3

LEG

LEG

HYPOTENUSE

Interior Angles Exterior Angles

Triangle Sum TheoremTriangle Sum TheoremThe measures of the three interior angles

in a triangle add up to be 180º.

x°

y° z°

x + y + z = 180°

54°

67°

R

S T

m R + m S + m T = 180º 54º + 67º + m T = 180º

121º + m T = 180º

m T = 59º

Find in RST.m T

85° x°55°

y°

A

B

C

D

E m D + m DCE + m E = 180º55º + 85º + y = 180º

140º + y = 180º

y = 40º

Find the value of each variable in DCE

Find the value of each variable.

x = 50º

x°

x° 43°

57°

Find the value of each variable.

x = 22º

(6x – 7)°43°55°

28°

(40 + y)°

y = 57º

Find the value of each variable.

x = 65º

62°

50°

50°

53°

x°

The measure of the exterior angle is equal to the sum of two nonadjacent interior angles

1

2 3

m1+m2 =m3

Exterior Angle TheoremExterior Angle Theorem

x

43

3881

148

72

x76

Ex. 1: Find x.

A. B.