1. 1-4 Angles

2. EXAMPLE 3 Find angle measures 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 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 o o

3. 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

4. GUIDED PRACTICE for Example 3 Find the indicated angle measures. 3. Given that KLM is straight angle, find m KLN and m NLM. STEP 1 Write and solve an equation to find the value of x. Straight angle Substitute angle measures. Combine like terms. Subtract 2 from each side. Divide each side by 14. m KLM + m NLM= 180° (10x – 5)° + (4x +3)°= 180° 14x – 2 = 180 14x = 182 x = 13 SOLUTION

5. GUIDED PRACTICE for Example 3 STEP 2 Evaluate the given expressions when x = 13. m KLM = (10x – 5)° = (10 13 – 5)° = 125° m NLM = (4x + 3)° = (4 13 + 3)° = 55° ANSWER m KLM= 125°m NLM = 55°

6. GUIDED PRACTICE for Example 3 4. Given that EFG is a right angle, find m EFH and m HFG. STEP 1 Write and solve an equation to find the value of x. Substitute angle measures. Combine like terms. Subtract 3 from each side. Divide each side by 3. m EFG + m HFG m EFG = = 90° (2x + 2)° + (x +1)°= 90° 3x + 3 = 90 3x3x = 87 x = 29 EFG is a right angle SOLUTION

7. GUIDED PRACTICE for Example 3 STEP 2 Evaluate the given expressions when x = 29. m EFH = (2x + 2)° = (2 29 +2)° = 60° m HFG = (x + 1)° = (29 + 1)° = 30° ANSWER m EFG= 60°m HFG = 30°

8. EXAMPLE 4 Identify congruent angles The photograph shows some of the angles formed by the ropes in a trapeze apparatus. Identify the congruent angles. If m DEG = 157°, what is m GKL? Trapeze SOLUTION There are two pairs of congruent angles: DEF JKL and DEG GKL. ~~ Because  DEG GKL, DEG = m GKL. So, m GKL = 157°. ~

9. GUIDED PRACTICE for Example 4 5. Identify all pairs of congruent angles in the diagram. Use the diagram shown below. There are two pairs of Congruent angles in the diagram. T S and P Q. ~~ SOLUTION

10. GUIDED PRACTICE for Example 4 6. In the diagram, m PQR = 130, m QRS = 84, and m TSR = 121. Find the other angle measures in the diagram. o o o PTS TSR ~ = 121° ~ QRSQPT= 84° Use the diagram shown at the right. SOLUTION Congruent angles

11. SOLUTION EXAMPLE 5 Double an angle measure In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ. o By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~ So, m XYW = m WYZ, and you can write M XYZ = m XYW + m WYZ = 18° + 18° = 36°.

12. GUIDED PRACTICE for Example 5 7. Angle MNP is a straight angle, and NQ bisects MNP. Draw MNP And NQ. Use arcs to mark the congruent angles in your diagram, and give the angle measures of these congruent angles. SOLUTION

13. GUIDED PRACTICE for Example 5 The solution is = 90° m MNQ = m PNQ m MNQ + m PNQ= 180° m MNQ + m PNQ m MNQ + m MNQ= 180° m MNQ2= 180°m MNQ= 90° Angle addition postulate Straight angle m MNQ = m PNQ Add Divided each side by 2

14. EXAMPLE 1 Name angles Name the three angles in the diagram. WXY, or YXW YXZ, or ZXY WXZ, or ZXW You should not name any of these angles X because all three angles have X as their vertex.

15. EXAMPLE 2 Measure and classify angles Use the diagram to find the measure of the indicated angle. Then classify the angle. a. KHJ b. GHK c. GHJ d. GHL SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

16. EXAMPLE 2 Measure and classify angles a. HJ is lined up with the 0 on the inner scale of the protractor. HK passes through 55 on the inner scale. So, m KHJ = 55. It is an acute angle. o o o b. HG is lined up with the 0 on the outer scale and HK passes through 125 on the outer scale. So, m GHK = 125. It is an obtuse angle. o o o c. m GHJ = 180. It is a straight angle. o d. m GHL= 90. It is a right angle. o

17. GUIDED PRACTICE for Examples 1and 2 1. Name all the angles in the diagram at the right.Which angle is a right angle? PQR, PQS, RQS. PQS is a right angle. ANSWER

18. GUIDED PRACTICE for Examples 1and 2 2. Draw a pair of opposite rays. What type of angle do the rays form? ANSWER Straight Angle