Sec 3.3 Angle Addition Postulate & Angle Bisector

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.

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. 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:

Steps 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° 65° C O

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.

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 = 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.

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 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 mÐBQC = 35 mÐCQD = 83 mÐBQD = 118 Algebra Connection Slide 5

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

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

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

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 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

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°.

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.

EXAMPLE 3 Find angle measures 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° So, m LKM = 56° and m MKN = 89°. ANSWER Back to Notes.

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

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

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

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

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°

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

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

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°

Constructing an angle bisector Folding

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

* 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

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 · 14 + 1)° = (84 + 1)° = 85° CHECK 31

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