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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14.
1
44°
15. 2
1
146°
16.
1
65°
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