Sec 3.3 Angle Addition Postulate & Angle Bisector

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° C 65° O

8. 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.

9. 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.

10. 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

11. 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

12. 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

13. 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

14. Find the indicated angle measures. 3. Given that KLMis a straight angle, find mKLN andm NLM. ANSWER 125°, 55° for Example 3 GUIDED PRACTICE

15. 4. Given that EFGis a right angle, find mEFH andm HFG. ANSWER 60°, 30° for Example 3 GUIDED PRACTICE

16. 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.

17. 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.

18. 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.

19. 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.

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? E H F G

22. Example 1 SOLUTION mABD BDbisects ABC. = Simplify. ANSWER Find Angle Measures BD bisectsABC, andmABC = 110°. Find mABDand mDBC. 1 1 2 2 (mABC) Substitute110°formABC. = (110°) = 55° ABD andDBCare congruent, somDBC= mABD. So, mABD = 55°, and mDBC = 55°.

23. Example 2 MP bisectsLMN, andmLMP =46°. b. Determine whether LMN is acute, right, obtuse, or straight. Explain. SOLUTION a. MP bisectsLMN, somLMP = mPMN . b. LMN is obtuse because its measure is between 90° and 180°. Find Angle Measures and Classify an Angle a. Find mPMNandmLMN. You know thatm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°, andmLMN = 92°

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

25. Checkpoint QSbisects PQR. Find mSQPand mPQR. 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.

26. Example 3 SOLUTION mDAB = 2(mABC) ACbisects DAB. Substitute45°formBAC. mBCD = CAbisects BCD. Real Life In the kite, DAB is bisected AC,and BCD is bisected by CA.Find mDAB and mBCD. 2(mACB) Simplify. 2(45°) = = 2(27°) = = 90° 54° Substitute27°formACB. Simplify. The measure of DABis 90°, and the measure of BCDis 54°. ANSWER

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

28. Constructing an angle bisector Folding

29. 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. A X E O Y B

30. 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. Example 4 RQ bisects PRS.Find the value of x. RQbisects PRS. Use Algebra with Angle Measures SOLUTION mQRS Substitute givenmeasures. (6x + 1)° Subtract 1 from each side. mPRQ = 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. mPRQ = (6x + 1)° = (6 · 14 + 1)° = (84 + 1)° = 85°

32. 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.