Put everything away and get out a piece

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 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 0 o on the inner scale of the o protractor. HK passes through 55 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 1 and 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 1 and 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 = (2 x + 10)o + (4 x – 3) 145 = 6 x + 7 138 = 6 x 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 = (2 x + 10)° = (2 23 + 10)° = 56° m MKN = (4 x – 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 = 130 o, m QRS = 84, o and m TSR = 121 o. 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, and SOLUTION By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects you know that XYW ~ WYZ. So, m m XYW = m XYZ = m WYZ, and you can write XYW + m WYZ = 18° + 18° = 36°. XYZ

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. ANSWER 90°

Homework! • HW: 29: 23 -47 odd, 53, 54