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Sec 3.3 Angle Addition Postulate & Angle Bisector

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Sec 3.3 Angle Addition Postulate & Angle Bisector

- Find the measure of an angle by using Angle Addition Postulate.
- Find the measure of an angle by using definition of Angle Bisector.

Angle Addition Postulate

First, let’s recall some previous information from last week….

We discussed the Segment Addition Postulate, which stated that we could add the lengths of adjacent segments together to get the length of an entire segment.

For example:

JK + KL = JL

If you know that JK = 7 and KL = 4, then you can conclude that JL = 11.

The Angle Addition Postulate is very similar, yet applies to angles. It allows us to add the measures of adjacent angles together to find the measure of a bigger angle…

J

K

L

If Q is between P and R, then

PQ + QR = PR.

If PQ +QR = PR, then Q is between P and R.

2x

4x + 6

R

P

Q

PQ = 2x QR = 4x + 6 PR = 60

Use the Segment Addition Postulate find the measure of PQ and QR.

Step 1:

PQ + QR = PR (Segment Addition)

2x + 4x + 6 = 60

6x + 6 = 60

6x = 54

x =9

PQ = 2x = 2(9) = 18

QR =4x + 6 = 4(9) + 6 = 42

Step 2:

Step 3:

Step 4:

- Draw and label the Line Segment.
- Set up the Segment Addition/Congruence Postulate.
- Set up/Solve equation.
- Calculate each of the line segments.

Angle Addition Postulate

Slide 2

If B lies on the interior of ÐAOC,

then mÐAOB + mÐBOC = mÐAOC.

B

A

mÐAOC = 115°

50°

C

65°

O

D

Example 1:

Example 2:

Slide 3

G

114°

K

134°

46°

A

B

C

95°

19°

This is a special example, because the two adjacent angles together create a straight angle.

Predict what mÐABD + mÐDBC equals.

ÐABC is a straight angle, therefore mÐABC = 180.

mÐABD + mÐDBC = mÐABC

mÐABD + mÐDBC = 180

So, if mÐABD = 134,

then mÐDBC = ______

H

J

Given: mÐGHK = 95

mÐGHJ = 114.

Find: mÐKHJ.

The Angle Addition Postulate tells us:

mÐGHK + mÐKHJ = mÐGHJ

95 + mÐKHJ = 114

mÐKHJ = 19.

Plug in what you know.

46

Solve.

Given:

mÐRSV = x + 5

mÐVST = 3x - 9

mÐRST = 68

Find x.

Algebra Connection

Slide 4

R

V

Extension: Now that you know x = 18, find mÐRSV and mÐVST.

mÐRSV = x + 5

mÐRSV = 18 + 5 = 23

mÐVST = 3x - 9

mÐVST = 3(18) – 9 = 45

Check:

mÐRSV + mÐVST = mÐRST

23+ 45 =68

S

T

Set up an equation using the Angle Addition Postulate.

mÐRSV + mÐVST = mÐRST

x + 5 + 3x – 9= 68

4x- 4 = 68

4x = 72

x = 18

Plug in what you know.

Solve.

mÐBQC = x – 7 mÐCQD = 2x – 1 mÐBQD = 2x + 34

Find x, mÐBQC, mÐCQD, mÐBQD.

C

B

mÐBQC = x – 7

mÐBQC = 42 – 7 = 35

mÐCQD = 2x – 1

mÐCQD = 2(42) – 1 = 83

mÐBQD = 2x + 34

mÐBQD = 2(42) + 34 = 118

Check:

mÐBQC + mÐCQD = mÐBQD

35+83 = 118

Q

D

mÐBQC + mÐCQD = mÐBQD

x – 7 + 2x – 1 = 2x + 34

3x – 8 = 2x + 34

x – 8 = 34

x = 42

x = 42

mÐCQD = 83

Algebra Connection

Slide 5

mÐBQC = 35

mÐBQD = 118

o

ALGEBRAGiven that m LKN =145 , find m LKM andm MKN.

So, m LKM = 56°and m MKN = 89°.

ANSWER

Animated Solution

EXAMPLE 3

Find angle measures

o

ALGEBRAGiven that m LKN =145 , find m LKM andm MKN.

So, m LKM = 56°and m MKN = 89°.

ANSWER

Animated Solution

EXAMPLE 3

Find angle measures

o

ALGEBRAGiven that m LKN =145 , find m LKM andm MKN.

So, m LKM = 56°and m MKN = 89°.

ANSWER

Animated Solution

EXAMPLE 3

Find angle measures

Find the indicated angle measures.

3. Given that KLMis a straight angle, find mKLN andm NLM.

ANSWER

125°, 55°

for Example 3

GUIDED PRACTICE

4. Given that EFGis a right angle, find mEFH andm HFG.

ANSWER

60°, 30°

for Example 3

GUIDED PRACTICE

ANSWER

T and S, P and R.

m T = 121°,

m P = 84°

ANSWER

Congruent Angles

Two angles are congruent if they have the same measure.

Congruent angles in a diagram are marked by matching arcs at the vertices .

Identify all pairs of congruent angles in the diagram.

In the diagram, m∠Q = 130° , m∠R = 84°, and m∠ S = 121° . Find the other angle measures in the diagram.

In the diagram at the right, YWbisectsXYZ, and mXYW = 18. Find m XYZ.

o

m XYZ = m XYW + m WYZ = 18° + 18° = 36°.

Angle Bisecotrs

An angle bisector is a ray that divides an angle into two congruent angles.

o

ALGEBRAGiven that m LKN =145 , find m LKM andm MKN.

STEP 1

Write and solve an equation to find the value of x.

mLKN = m LKM + mMKN

o

o

o

145 = (2x + 10)+ (4x – 3)

EXAMPLE 3

Animated Solution – Click to see steps and reasons.

SOLUTION

Angle Addition Postulate

Substitute angle measures.

145 = 6x + 7

Combine like terms.

138 = 6x

Subtract 7 from each side.

23 = x

Divide each side by 6.

STEP 2

Evaluate the given expressions when x = 23.

mLKM = (2x+ 10)° = (2 23+ 10)° = 56°

mMKN = (4x– 3)° = (4 23– 3)° = 89°

So, m LKM = 56°and m MKN = 89°.

ANSWER

EXAMPLE 3

Find angle measures

Back to Notes.

- A ray that divides an angle into 2 congruent adjacent angles.
BD is an angle bisector.bisector of <ABC.

A

D

B

C

E

H

F

G

Example 1

SOLUTION

mABD

BDbisects ABC.

=

Simplify.

ANSWER

Find Angle Measures

BD bisectsABC, andmABC = 110°.

Find mABDand mDBC.

1

1

2

2

(mABC)

Substitute110°formABC.

=

(110°)

=

55°

ABD andDBCare congruent, somDBC= mABD.

So, mABD = 55°, and mDBC = 55°.

Example 2

MP

bisectsLMN, andmLMP =46°.

b.

Determine whether LMN is acute, right, obtuse,

or straight. Explain.

SOLUTION

a.

MP

bisectsLMN, somLMP = mPMN .

b.

LMN is obtuse because its measure is between 90° and 180°.

Find Angle Measures and Classify an Angle

a.

Find mPMNandmLMN.

You know thatmLMP = 46°. Therefore, mPMN = 46°.

The measure ofLMN is twice the measure of LMP.

mLMN =2(mLMP) = 2(46°) = 92°

So, mPMN = 46°, andmLMN = 92°

Checkpoint

HK bisects GHJ. Find mGHK and mKHJ.

ANSWER

ANSWER

ANSWER

26°; 26°

80.5°; 80.5°

45°; 45°

Find Angle Measures

1.

2.

3.

Checkpoint

QSbisects PQR. Find mSQPand mPQR. Then determine whether PQRis acute, right, obtuse, or straight.

ANSWER

ANSWER

29°; 58°; acute

45°; 90°; right

ANSWER

60°; 120°; obtuse

Find Angle Measures and Classify an Angle

4.

5.

6.

Example 3

SOLUTION

mDAB

=

2(mABC)

ACbisects DAB.

Substitute45°formBAC.

mBCD

=

CAbisects BCD.

Real Life

In the kite, DAB is bisected AC,and BCD is bisected by CA.Find mDAB and mBCD.

2(mACB)

Simplify.

2(45°)

=

=

2(27°)

=

=

90°

54°

Substitute27°formACB.

Simplify.

The measure of DABis 90°, and the measure of BCDis 54°.

ANSWER

Checkpoint

7.

KM bisects JKL.

Find mJKM and mMKL.

ANSWER

ANSWER

48°; 48°

60°; 120°

8.

UVbisects WUT.

Find mWUVand mWUT.

Real Life

Folding

- Using the vertex O as a center, draw an arc to meet the arms of the angle (at X and Y).
- Using X as a center and the same radius, draw a new arc.
- Using Y as center and the same radius, draw an overlapping arc.
- Mark the point where the arcs meet.
- The bisector is the line from O to this point.

A

X

E

O

Y

B

* If they are congruent, set them equal to each other, then solve!

x+40o

x+40=3x-20

40=2x-20

60=2x

30=x

3x-20o

Example 4

RQ bisects PRS.Find the value of x.

RQbisects PRS.

Use Algebra with Angle Measures

SOLUTION

mQRS

Substitute givenmeasures.

(6x + 1)°

Subtract 1 from each side.

mPRQ

=

6x = 84

6x + 1 – 1

Simplify.

85°

=

Divide each side by 6.

=

85 – 1

6x

84

––

––

Simplify.

x = 14

=

6

6

CHECK

You can check your answer by substituting 14

for x.

mPRQ = (6x + 1)° = (6 · 14 + 1)° = (84 + 1)° = 85°

Checkpoint

BD bisects ABC. Find the value of x.

ANSWER

43

3

ANSWER

Use Algebra with Angle Measures

55 = x + 12

X =43

9.

9x = 8x + 3

x = 3

10.