1. Sec 3.3 Angle Addition Postulate & Angle Bisector

2. Objective: What we’ll learn…
Find the measure of an angle by using Angle Addition Postulate.
Find the measure of an angle by using definition of Angle Bisector.

3. 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

4. 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. 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.

5. 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:

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

7. Angle Addition Postulate
Slide 2
If B lies on the interior of ÐAOC,
then mÐAOB + mÐBOC = mÐAOC. B A
mÐAOC = 115°
50°
65° C O

8. A B C D G K H J 134° 46° 46 Given: mÐGHK = 95 mÐGHJ = 114.
Example 1:
Example 2:
Slide 3 G
114° K
46°
95°
19° H
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 = ______ 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.

9. Set up an equation using the Angle Addition Postulate.
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 = = 23
mÐVST = 3x - 9
mÐVST = 3(18) – 9 = 45
Check:
mÐRSV + mÐVST = mÐRST
= 68 S T
Set up an equation using the Angle Addition Postulate.
mÐRSV + mÐVST = mÐRST
x x – 9 = 68
4x- 4 = 68
4x = 72
x = 18
Plug in what you know.
Solve.

10. x – 7 + 2x – 1 = 2x + 34 3x – 8 = 2x + 34 x – 8 = 34 x = 42 x = 42 C B
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
= 118 Q D
mÐBQC + mÐCQD = mÐBQD
x – 7 + 2x – 1 = 2x + 34
3x – 8 = 2x + 34
x – 8 = 34
x = 42
x = 42
mÐBQC = 35
mÐCQD = 83
mÐBQD = 118
Algebra Connection
Slide 5

11. ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
Animated Solution
EXAMPLE 3
Find angle measures o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
So, m LKM = 56° and m MKN = 89°.
ANSWER

12. ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
Animated Solution
EXAMPLE 3
Find angle measures o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
So, m LKM = 56° and m MKN = 89°.
ANSWER

13. ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
Animated Solution
EXAMPLE 3
Find angle measures o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
So, m LKM = 56° and m MKN = 89°.
ANSWER

14. GUIDED PRACTICE
for Example 3
Find the indicated angle measures.
Given that KLM is a straight angle, find m KLN and m NLM.
ANSWER
125°, 55°

15. GUIDED PRACTICE
for Example 3
4. Given that EFG is a right angle, find m EFH and m HFG.
ANSWER
60°, 30°

16. 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.
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

17. 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.
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°.

18. Angle Addition Postulate
EXAMPLE 3
Animated Solution – Click to see steps and reasons. o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m LKN = m LKM + m MKN
Angle Addition Postulate
145 = (2x + 10) + (4x – 3) o
Substitute angle measures.
145 = 6x + 7
Combine like terms.
138 = 6x
Subtract 7 from each side.
23 = x
Divide each side by 6.

19. EXAMPLE 3
Find angle measures
STEP 2
Evaluate the given expressions when x = 23.
m LKM = (2x + 10)° = ( )° = 56°
m MKN = (4x – 3)° = (4 23 – 3)° = 89°
So, m LKM = 56° and m MKN = 89°.
ANSWER
Back to Notes.

20. 3.3 Angle Bisector
A ray that divides an angle into 2 congruent adjacent angles.
BD is an angle bisector bisector of <ABC. A D B C

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

22. Find Angle Measures BD bisects ABC. Substitute 110° for mABC.
Example 1
Find Angle Measures
BD bisects ABC, and mABC = 110°.
Find mABD and mDBC.
SOLUTION 2 1
(mABC)
mABD =
BD bisects ABC. 2 1 =
(110°)
Substitute 110° for mABC. =
55°
Simplify.
ABD and DBC are congruent, so mDBC = mABD.
ANSWER
So, mABD = 55°, and mDBC = 55°. 22

23. Find Angle Measures and Classify an Angle
Example 2
Find Angle Measures and Classify an Angle
bisects LMN, and mLMP = 46°. MP
Find mPMN and mLMN. a.
Determine whether LMN is acute, right, obtuse,
or straight. Explain. b.
SOLUTION a.
bisects LMN, so mLMP = mPMN . MP
You know that mLMP = 46°. Therefore, mPMN = 46°.
The measure of LMN is twice the measure of LMP.
mLMN = 2(mLMP) = 2(46°) = 92°
So, mPMN = 46°, and mLMN = 92°
LMN is obtuse because its measure is between 90° and 180°. b. 23

24. Find Angle Measures Checkpoint HK bisects GHJ. Find mGHK and mKHJ.1.
ANSWER
26°; 26° 2.
ANSWER
45°; 45° 3.
ANSWER
80.5°; 80.5°

25. Find Angle Measures and Classify an Angle
Checkpoint
Find Angle Measures and Classify an Angle
QS bisects PQR. Find mSQP and mPQR. Then determine whether PQR is acute, right, obtuse, or straight. 4.
ANSWER
29°; 58°; acute 5.
ANSWER
45°; 90°; right 6.
ANSWER
60°; 120°; obtuse

26. Real Life AC bisects DAB. Substitute 45° for mBAC. Simplify.
Example 3
Real Life
In the kite, DAB is bisected AC, and BCD is bisected by CA. Find mDAB and mBCD.
SOLUTION
2(mABC)
mDAB =
AC bisects DAB. =
2(45°)
Substitute 45° for mBAC. =
90°
Simplify.
2(mACB)
mBCD =
CA bisects BCD.
Substitute 27° for mACB. =
2(27°) =
54°
Simplify.
The measure of DAB is 90°, and the measure of BCD is 54°.
ANSWER 26

27. Real Life Checkpoint 7. KM bisects JKL. Find mJKM and mMKL. ANSWER
48°; 48° 8.
UV bisects WUT.
Find mWUV and mWUT.
ANSWER
60°; 120°

28. Constructing an angle bisector
Folding

29. Construct the bisector of an angle using 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. A X E O Y B

30. * If they are congruent, set them equal to each other, then solve!
Solve for x.
* 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

31. Substitute given measures.
Example 4
Use Algebra with Angle Measures
RQ bisects PRS. Find the value of x.
SOLUTION
mPRQ =
mQRS
RQ bisects PRS.
Substitute given measures. =
85°
(6x + 1)°
Subtract 1 from each side. =
85 – 1
6x + 1 – 1
Simplify.
6x = 84
Divide each side by 6. 6x 6 –– = 84
Simplify.
x = 14
You can check your answer by substituting 14
for x.
mPRQ = (6x + 1)° = (6 · )° = (84 + 1)° = 85°
CHECK 31

32. Use Algebra with Angle Measures
Checkpoint
Use Algebra with Angle Measures
BD bisects ABC. Find the value of x. 9.
55 = x + 12
X =43
ANSWER 43
10.
9x = 8x + 3
x = 3
ANSWER 3