Angle Addition Postulate Angle Bisector Steps 1 Draw

Angle Addition Postulate & Angle Bisector

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.

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 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: J K JK + KL = JL L 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…

If Q is between P and R, then PQ + QR = PR. If PQ +QR = PR, then Q is between P and R. 2 x P 4 x + 6 R Q PQ = 2 x QR = 4 x + 6 PR = 60 Use the Segment Addition Postulate find the measure of PQ and QR.

Step 1: Step 2: Step 3: Step 4: PQ + QR = PR (Segment Addition) 2 x + 4 x + 6 = 60 6 x = 54 x =9 PQ = 2 x = 2(9) = 18 QR =4 x + 6 = 4(9) + 6 = 42

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

Example 1: G D Example 2: 114° K 95° 19° H Given: mÐGHK = 95 mÐGHJ = 114. Find: mÐKHJ. J 134° A Slide 3 46° B C 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. The Angle Addition Postulate tells us: mÐABD + mÐDBC = mÐABC mÐGHK + mÐKHJ = mÐGHJ 95 + mÐKHJ = 114 mÐKHJ = 19. Plug in what you know. Solve. mÐABD + mÐDBC = 180 So, if mÐABD = 134, 46 then mÐDBC = ______

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

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

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

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

Angle Bisecotrs An angle bisector is a ray that divides an angle into two congruent angles. In the diagram at the right, YW bisects Find m XYZ. o m XYZ = m XYW + m XYZ, and m WYZ = 18° + 18° = 36°. XYW = 18.

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

Checkpoint Find Angle Measures HK bisects GHJ. Find m GHK and m KHJ. 1. ANSWER 26°; 26° ANSWER 45°; 45° ANSWER 80. 5°; 80. 5° 2. 3.

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

Solve for x. * If they are congruent, set them equal to each other, then solve! o 40 x+ x+40=3 x-20 o 3 x-20 40=2 x-20 60=2 x 30=x

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

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