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Put everything away and get out a piece of paper. Make sure you put your name and period on the paper. • Find the length of segment ST. Round to the nearest tenth. Definition Angle: two different rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex. 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. Protractor Postulate: The measure of angle AOB is the absolute value of the difference between the real numbers for OA and OB. A O B Classifying Angles: Angles can be classified as acute, right, obtuse, and straight. 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. SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use. GHJ d. GHL EXAMPLE 2 Measure and classify angles a. HJ is lined up with the 0o on the inner scale of the protractor. HK passes through 55o on the inner o scale. So, m KHJ = 55 . It is an acute angle. b. HG is lined up with the 0 on the outer scale and o HK passes through 125 on the outer scale. So, m o GHK = 125 . It is an obtuse angle. c. m GHJ = 180.o It is a straight angle. o o d. m GHL= 90. It is a right angle. GUIDED PRACTICE 1. for Examples 1and 2 Name all the angles in the diagram. Which angle is a right angle? ANSWER PQR , PQS, RQS ; PQS is a right angle . GUIDED PRACTICE 2. for Examples 1and 2 Draw a pair of opposite rays. What type of angle do the rays form? ANSWER Straight Angle Homework! HW: P.28: 1-21 odd Angle Addition Postulate EXAMPLE 3 Find angle measures ALGEBRA Given that m and m MKN. o LKN =145 , find m LKM SOLUTION STEP 1 Write and solve an equation to find the value of x. m LKN = m LKM + m MKN o o 145 = (2x + 10)o + (4x – 3) 145 = 6x + 7 138 = 6x 23 = x Angle Addition Postulate Substitute angle measures. Combine like terms. Subtract 7 from each side. 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° ANSWER So, m LKM = 56° and m MKN = 89°. GUIDED PRACTICE for Example 3 Find the indicated angle measures. 3. Given that KLM is a straight angle, find m and m NLM. ANSWER 125°, 55° KLN GUIDED PRACTICE 4. Given that and m HFG. for Example 3 EFG is a right angle, find m ANSWER 60°, 30° EFH Definition Congruent Angles: Two angles are contruent angles if they have the same measure. EXAMPLE 4 Identify congruent angles Trapeze 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? SOLUTION There are two pairs of congruent angles: DEF ~ JKL and DEG ~ GKL. Because DEG~ GKL, DEG = m GKL. So, m GKL = 157°. GUIDED PRACTICE for Example 4 Use the diagram shown. 5. Identify all pairs of congruent angles in the diagram. ANSWER T and S, P and R. GUIDED PRACTICE for Example 4 Use the diagram shown. 6. In the diagram, m PQR = 130o, m QRS = 84,o and m TSR = 121o . Find the other angle measures in the diagram. ANSWER m PTS = 121°, m QPT = 84° Definition Angle Bisector: A ray that divides an angle into two angles that are congruent EXAMPLE 5 Double an angle measure In the diagram at the right, YW bisects o m XYW = 18. Find m XYZ. XYZ, and SOLUTION By the Angle Addition Postulate, XYZ = m XYW + m WYZ. Because YW bisects you know that XYW ~ WYZ. So, m m XYW = m WYZ, and you can write XYZ = m XYW + m WYZ = 18° + 18° = 36°. m XYZ GUIDED PRACTICE 7. for Example 5 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. ANSWER 90° Homework! • HW: 29: 23-47 odd, 53, 54