Lesson 6.4 Interior and Exterior Angles

o find the v

B

A C58° 41°

m!ABC ! m!ACB ! m!BAC " 180° m!ABC ! 41° ! 58° " 180° m!ABC ! 99° " 180° m!ABC ! 99° # 99° " 180° # 99° m!ABC " 81°

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

1.

37°

56°

x°

x ! ! "

x ! "

x ! # " #

x "

o find the v

2.

65°87°

x°

! sum of triangleSubstitute.Simplify.Subtract 99° from both sides.

Simplify.

Substitute.

! sum of triangle

Subtract from both sides.

Simplify.

Example

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o find the v

3.

20° 15°x°

4. 118°

42°

x°

o find the unkno

Triangle ABC is an isosceles triangle. Find the measure of !ABC.

40°

x°

B

A

C

m!ABC ! m!ACB ! x° m!ABC " m!ACB " m!BAC ! 180° x° " x° " 40° ! 180° 2x° " 40° ! 180° 2x° " 40° # 40° ! 180° # 40° 2x ! 140

2x2 ! 140

2 x ! 70

Example

In an isosceles triangle, the measures of the angles opposite the equal sides are congruent.

Isosceles triangle

!sum of triangleSubstitute.Simplify.Subtract 40° from both sides.

Divide both sides by 2.

Simplify.

Simplify.

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24 Chapter 6 Lesson 6.4

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5. Triangle PQR is an isosceles triangle, and m!RPQ is 63°. Find the value of x.

63°

x°

P

Q

R

m!QRP ! m!QPR !

m!PQR " m!PRQ " m!QPR !

x " " !

x " !

x " # ! #

x !

6. Triangle ABC is an isosceles triangle.

x°

A

B C

7. Triangle XYZ is an isosceles triangle, and m!YXZ is 60°.

x°

60°

Y Z

X

Isosceles triangle

!sum of triangle

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8. Triangle PQR is an isosceles triangle, and m!QPR is 20°.

x°

20°

Q

P

R

o find the v

a)

x°

48°

65°

b) x°

62°121°

Example

Ext. ! of trianglex° ! 65° " 48° x ! 113

9.

x° 66°

130°

Ext. ! of triangle x° " 62° ! 121° x° " 62° # 62° ! 121° # 62° x ! 59

Subtract 62° from both sides.Simplify.

Ext. ! of triangle x° " !

x° " # ! #

x !

Subtract 66° from both sides.

Simplify.

The exterior angle of a triangle is always equal to the sum of the other two interior angles of the triangle.

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26 Chapter 6 Lesson 6.4

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o find the v

10. x°

50°

60°

11. x°

132°

50°

12.

x°

129°

79°

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12

70°

R

P Q

RP ! RQ So, m!PQR ! m!RPQ ! m!1

m!1 " m!RQP " m!PRQ ! 180° m!1 " m!1 " 70° ! 180° 2 # m!1 " 70° ! 180° 2 # m!1 " 70° $ 70° ! 180° $ 70° 2m!1 ! 110°

2m!12 ! 110°

2 m!1 ! 55°

m!2 ! 70° " m!1 m!2 ! 70° " 55° m!2 ! 125°

Example

! sum of triangle

Ext. ! of triangle

13. Triangle ABC is an isosceles triangle.

2

1130°

B

ACAdj. !s on a st. line

Ext. ! of triangle

AC ! AB

So, m!ACB ! m!ABC !

130° " m!ACB ! 180°

130° " ! 180°

130° " $ ! 180° $

m!2 !

m!1 " m!2 ! 130°

m!1 " ! 130°

m!1 " $ ! 130° $

m!1 !

Substitute m!RQP ! m!1.

Subtract 70° from both sides.

Substitute m!1 ! 55°.

Divide both sides by 2.

Simplify.

Simplify.

Simplify.

Simplify.

Subtract from both sides.

Substitiute m!2 !

.Subtract from both sides.

Simplify.

Simplify.

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28 Chapter 6 Lesson 6.4

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14.

1

44°

15. 2

1

146°

16.

1

65°

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