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Geometry Section 4-1BGeometry Section 4-1B

Angles Outside the TriangleAngles Outside the Triangle

Pg. 247Pg. 247

Be ready to grade 4-1ABe ready to grade 4-1AQuiz FridayQuiz Friday

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The last section of this chapter referred to angles inside of a triangle. Those are called _____________________.interior angles

X Y

Z

WX, Y, and Z

are interior angles.

Definition

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Exterior Angle:

An angle of a triangle that is formed by extending one of the sides of the triangle.

X Y

Z

W W is an exterior angle.

Definition

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Adjacent Interior Angle:

The angle that forms a linear pair with the exterior angle.

X Y

Z

W Z is an adjacent interior angle with W.

Definition

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Remote Interior Angle:

An angle NOT adjacent to the exterior angle.

X Y

Z

W X and Y are remote

interior angles to W.

Definition

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1, 2 and 3

4 and 5

52

3

4 61

a. Exterior Angles

b. Remote Interior Angles for 3

78

Note that 8 is an exterior angle, but 7 is not.

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9 is not an exterior angle. Why?

Try It:

Pg.248

Try It:

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m1 m2 m3 m4

83o 40o

84o

71o

148o

54o

1 2

3

4

57o 97o

32o 64o

55o 125o

180 – 83 = 97

180 – (83 + 40) = 57

180 – 148= 32

180 – (84 + 32) = 64

180 – (71 + 54) = 55

180 – 55= 125

Try It:

Pg.249

Try It:

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Exterior Angle Theorem:

The measure of an exterior angle of a triangle is equal to the sum of the measures of its

remote interior angles.

X Y

Z

W

mX + mY = mW.

Pg.249

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Exterior Angle Inequality Theorem:The measure of an exterior angle of a triangle

is greater than the measure either of its remote interior angles.

X Y

Z

W

A theorem that follows immediately FROM another is called a _____________.corollary

If we know that mX + mY = mW, then….

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Each angle of an equilateral triangle measures 60o.

60o

60o 60o

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Theorem

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The acute angles of a right triangle are complementary.

a

b

ma + mb = 90o

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Theorem

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What are the remote angles for 1?

123

4

1

2

3

4

4

2

31

3 and 4

2 and 4

2 and 3

#1 – 3

Pg.250

Exercises

13

Find m5. 5

8

6

7

m5 = m7 = (2x + 31)o

m6 = xo m8 = (4x - 19)o

x + 2x + 31 + 4x – 19 = 180

x = 24

180 – 24 =156o

#7

Pg.250

Exercises

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The brace JK is fastened to the wall and the shelf shown. Which angle has the greater measure, LJK or MKJ? Justify your answer.

#10

Pg.250

MKJ has the greater measure because of the Exterior Angle Inequality Theorem. LJK is a

remote interior angle for MKJ.

Exercises

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The measures of the acute angles of a right triangle are in the ratio of 1:2. What is the measure of the smallest angle of the triangle?

#11

Pg.250

We know that the acute angles of a right triangle are complementary therefore:

Exercises

1x + 2x = 90

x = 30o

Be sure to pay attention to which angle they are asking you for! Biggest or smallest?

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If the measures of the exterior angles of the acute angles of a right triangle are (6x + 23)o

and (4x + 17)o, find the measures of the acute

angles.

#12

Pg.251

Exercises

17

3

C

24 1DA B

If m1 = 18o, find m2.

#13

Pg.251

m2 = 180 – (90 + 18) = 72o

Exercises

18

3

C

24 1DA B

If m4 = 150o and m3 = 130o find mACD.

#14

Pg.251

Find the measure of CAD.

CAD = 180 – 150 = 30o.

ACD = 180 – (130 + 30) = 20o.

Exercises

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3

C

24 1DA B

If m1 = 20o, find m3.

#15

Pg.251

Find the measure of 2.

2 = 180 – (90 + 20) = 70o.

3 = 180 – 70= 110o.

Exercises

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A1 2

B

D

C

3

5

4E

If AC || BE, explain how the figure above shows that

m1 + m2 + m3 = 180o.

#25

Pg.252

By the properties of parallel lines, we know that m1 = m5 and m2 = m4.

Since m3 + mm5 = 180, we can use substitution to prove that m1 + m2 + m3 = 180o.

Exercises

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Homework: Practice 4-1B

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