Sec 3.3 Angle Addition Postulate & Angle Bisector

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.

Objective: What we’ll learn…

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

Postulate 2-2 Segment Addition Postulate

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

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

P Q R

2x 4x + 6

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

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

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 1:

Step 2:

Step 3:

Step 4:

Steps

1. Draw and label the Line Segment.2. Set up the Segment Addition/Congruence

Postulate.3. Set up/Solve equation.4. Calculate each of the line segments.

Angle Addition Postulate

5065

AB

CO

If B lies on the interior of AOC, then mAOB + mBOC = mAOC.

mAOC = 115

Slide 2

Example 1:

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

Predict what mABD + mDBC equals.

ABC is a straight angle, therefore mABC = 180.

mABD + mDBC = mABC

mABD + mDBC = 180

So, if mABD = 134, then mDBC = ______

A B C

D

134°

46

G

H J

K

Given: mGHK = 95 mGHJ = 114.

Find: mKHJ.

The Angle Addition Postulate tells us:mGHK + mKHJ = mGHJ

95 + mKHJ = 114

mKHJ = 19.

95

114

19

Plug in what you

know.

Solve.

46°

Example 2: Slide 3

Algebra Connection

Slide 4R

S T

V

Given:mRSV = x + 5mVST = 3x - 9mRST = 68

Find x.

mRSV + mVST = mRST

x + 5 + 3x – 9 = 68

4x- 4 = 68

4x = 72

x = 18

Set up an equation using the Angle Addition Postulate.

Plug in what you

know.Solve.

Extension: Now that you know x = 18, find mRSV and mVST.

mRSV = x + 5mRSV = 18 + 5 = 23

mVST = 3x - 9mVST = 3(18) – 9 = 45

Check:mRSV + mVST = mRST23 + 45 = 68

Algebra Connection

Slide 5

B

QD

CmBQC = x – 7 mCQD = 2x – 1 mBQD = 2x + 34

Find x, mBQC, mCQD, mBQD.

mBQC + mCQD = mBQD

3x – 8 = 2x + 34

x – 7 + 2x – 1 = 2x + 34

x – 8 = 34

x = 42

mBQC = 35

mCQD = 83

mBQD = 118

x = 42

mBQC = x – 7mBQC = 42 – 7 = 35

mCQD = 2x – 1mCQD = 2(42) – 1 = 83

mBQD = 2x + 34mBQD = 2(42) + 34 = 118

Check:mBQC + mCQD = mBQD35 + 83 = 118

EXAMPLE 3 Find angle measures

oALGEBRA Given that m LKN =145 , find m LKM and m MKN.

So, m LKM = 56° and m MKN = 89°.ANSWER







GUIDED PRACTICE for Example 3

Find the indicated angle measures.

3. Given that KLM is a straight angle, find m KLN and m NLM.

ANSWER 125°, 55°

GUIDED PRACTICE for Example 3

4. Given that EFG is a right angle, find m EFH and m HFG.

ANSWER 60°, 30°

Congruent Angles

Identify all pairs of congruent angles in the diagram.

T and S, P and R.ANSWER

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

m T = 121°, m P = 84°ANSWER

Two angles are congruent if they have the same measure.

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

Angle Bisecotrs

In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ. o

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

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

EXAMPLE 3 Animated Solution – Click to see steps and reasons.


SOLUTION

STEP 1

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

m LKN = m LKM + m MKN Angle Addition Postulate

Substitute angle measures.

145 = 6x + 7 Combine like terms.

Subtract 7 from each side.138 = 6x

Divide each side by 6.23 = x

145 = (2x + 10) + (4x – 3)o oo


STEP 2

Evaluate the given expressions when x = 23.

m LKM = (2x + 10)° = (2 23 + 10)° = 56°

m MKN = (4x – 3)° = (4 23 – 3)° = 89°


3.3 Angle Bisector3.3 Angle Bisector

• A ray that divides an angle into 2 congruent adjacent angles.

BD is an angle bisector. bisector of <ABC.B

A

C

D

Ex: If FH bisects <EFG & m<EFG=120o, what is m<EFH?

G

H

E

F

o602

120

oEFHm 60

Example 1 Find Angle Measures

2

1= (110°) Substitute 110° for mABC.

= 55° Simplify.

ABD and DBC are congruent, so mDBC = mABD.

ANSWER So, mABD = 55°, and mDBC = 55°.

SOLUTION

2

1(mABC)mABD = BD bisects ABC.

BD bisects ABC, and mABC = 110°.Find mABD and mDBC.

You know that mLMP = 46°. Therefore, mPMN = 46°.

The measure of LMN is twice the measure of LMP.

mLMN = 2(mLMP) = 2(46°) = 92°

So, mPMN = 46°, and mLMN = 92°

Example 2 Find Angle Measures and Classify an Angle

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

Find mPMN and mLMN.a.

Determine whether LMN is acute, right, obtuse, or straight. Explain.

b.

bisects LMN, and mLMP = 46°.MP

SOLUTION

a. bisects LMN, so mLMP = mPMN .MP

Checkpoint Find Angle Measures

ANSWER 26°; 26°

ANSWER 80.5°; 80.5°

ANSWER 45°; 45°

1.

2.

3.

HK bisects GHJ. Find mGHK and mKHJ.

Checkpoint Find Angle Measures and Classify an Angle

ANSWER 29°; 58°; acute

ANSWER 45°; 90°; right

ANSWER 60°; 120°; obtuse

5.

6.

4.

QS bisects PQR. Find mSQP and mPQR. Then determine whether PQR is acute, right, obtuse, or straight.

Example 3 Real Life

= 2(45°) Substitute 45° for mBAC.

= 90° Simplify.

Substitute 27° for mACB.= 2(27°)

= 54° Simplify.

The measure of DAB is 90°, and the measure of BCD is 54°.

ANSWER

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

2(mACB)mBCD = CA bisects BCD.

SOLUTION

2(mABC)mDAB = AC bisects DAB.

Checkpoint Real Life

ANSWER 48°; 48°

ANSWER 60°; 120°

7. KM bisects JKL.Find mJKM and mMKL.

8. UV bisects WUT.Find mWUV and mWUT.

Constructing an angle bisector

Construct the bisector of an angleusing a compass and straight edge

• 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.

Y

X

O

A

B

E

Solve for x.

x+40o

3x-20o

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

then solve!

x+40=3x-20

40=2x-20

60=2x

30=x

Example 4 Use Algebra with Angle Measures

Simplify.x = 14

Substitute given measures.= 85°(6x + 1)°

Subtract 1 from each side.= 85 – 16x + 1 – 1

Simplify.6x = 84

Divide each side by 6.6x

6–– =

84

6––

You can check your answer by substituting 14 for x. mPRQ = (6x + 1)° = (6 · 14 + 1)° = (84 + 1)° = 85°

CHECK

SOLUTION

mPRQ = mQRS RQ bisects PRS.

RQ bisects PRS. Find the value of x.

Checkpoint Use Algebra with Angle Measures

ANSWER 43

ANSWER 3

9.

10.

BD bisects ABC. Find the value of x.

55 = x + 12 X =43

9x = 8x + 3 x = 3

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Sec 3.3 Angle Addition Postulate & Angle Bisector