1.4

1. 1.4 Angles and Their Measures 1 2 GOAL GOAL Use Angle Postulates Classify angles as acute, right, obtuse, or straight. To solve real-life problems about angles, such as the field of vision of a horse wearing blinkers. Whatyou should learn Why you should learn it

2. 1.4 Angles and Their Measures USING ANGLE POSTULATES 1 GOAL C A B An _____ consists of two different rays that have the same initial point. angle The rays are the _____ of the angle and the initial point is the ______. sides vertex The three names for this angle are The sides are The vertex is point A.

3. 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 Xas their vertex. EXAMPLE 1 Name angles

4. M N O P should not be used to name any angle in the figure. Why not? 1. 2. All three angles have N as the vertex, so could mean any of the angles. 3. EXAMPLE 1 Extra Example 1 Name the angles in the figure. Click for the answers.

5. Measuring Angles The expression is read as “the ________ of angle A.” measure IMPORTANT!!! Note the difference in notation between an angle and its measure. Always use the correct notation!!! The tool used to measure angles is called a _________. protractor The units used to measure angles are called _______, and the symbol for them is a _. degrees

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

7. o a. HJ is lined up with the 0 on the inner scale of the protractor. HKpasses through 55 on the inner scale. So, mKHJ = 55 . It is an acute angle. o o o b. HG is lined up with the 0 on the outer scale and HKpasses through 125 on the outer scale. So, mGHK = 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. EXAMPLE 2 Measure and classify angles

8. Since we say that the angles are _________. R M Y Incorrect: Q T and S Measuring Angles Let’s measure some angles. congruent Remember: Angles are congruent, measures are equal. Correct:

9. MEASURES ARE EQUAL mBAC = mDEF ANGLES ARE CONGRUENT BAC  DEF TAKE NOTE! “is congruent to” “is equal to” Note that there is an m in front when you say equal to; whereas the congruency symbol  ; you would say congruent to. (no m’s in front of the angle symbols).

10. For any point A on one side of , can be matched one to one with the real numbers from 0 to ___. The absolute value of the difference between the real numbers for is the ________ of Check your understanding of the Protractor Postulate by finding Use either scale on the protractor to find it, but use the same one for both rays. A B or O PROTRACTOR POSTULATE 180 measure Click to see two solutions.

11. exterior A B interior Z To understand the next postulate, you must understand some vocabulary: A point that is between points that lie on each side of an angle is in the _______ of the angle. A point that is not on an angle or in its interior is in the ________ of the angle. interior exterior In the above diagram, A is in the interior of the angle and B is in the exterior of the angle.

12. If P is in the interior of then R P S T EXAMPLE 2 ANGLE ADDITION POSTULATE Does it make sense? Study the figure to be sure you understand before going on!

13. Extra Example 2 The backyard of a house is illuminated by a light fixture that has two bulbs. Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs? Click for a picture. What is the measure of the angle that is shaded green? 120° Click for the solution. 120° 120° - 35° = 85° 85° Do you see the application of the Angle Addition Postulate? 35°

14. D • Name the angles in the figure. Click for the solution. C F E • In the figure above, and Find the measure of Click for the solution. Checkpoint

15. 1.4 Angles and Their Measures 2 GOAL CLASSIFYING ANGLES We can classify all angles by their measure as follows: Angles with measures between 0 and 90 degrees: _____ 90 degree angles: _____ Angles with measures between 90 and 180 degrees: ______ 180 degree angles: _______ acute right obtuse straight

16. A Important!!! Anytime this box appears at the vertex of an angle, it means that the angle is a right angle. You MUST use it when you want to show a right angle. acute: right: A obtuse: A straight: A EXAMPLE 3 Can you match each angle with its description? Click to check your answer.

17. Classifying Angles in a Coordinate Plane • Plot the points L(-4,2), M(-1,-1), N(2,2), Q(4,-1), and P(2,-4). Then measure and classify the following angles as acute, right, obtuse, or straight. • LMN • LMP • NMQ • LMQ

18. Solution: • Begin by plotting the points. Then use a protractor to measure each angle.

19. Solution: • Begin by plotting the points. Then use a protractor to measure each angle.

20. Plot the points A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2). Then measure and classify the following angles as acute, right, obtuse, or straight. Click for each answer. a. b. c. d. Extra Example 3 45°, acute 90°, right 180°, straight 135°, obtuse Reminder: Be sure you understand the material before going on to the next slide. If you need to review, do so NOW!

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

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

23. o ALGEBRAGiven that m LKN =145 , find m LKM andm MKN. STEP 1 Write and solve an equation to find the value of x. mLKN = m LKM + mMKN o o o 145 = (2x + 10)+ (4x – 3) EXAMPLE 3 Find angle measures SOLUTION Angle Addition Postulate Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6.

24. STEP 2 Evaluate the given expressions when x = 23. mLKM = (2x+ 10)° = (2 23+ 10)° = 56° mMKN = (4x– 3)° = (4 23– 3)° = 89° So, m LKM = 56°and m MKN = 89°. ANSWER EXAMPLE 3 Find angle measures

25. Find the indicated angle measures. 3. Given that KLMis a straight angle, find mKLN andm NLM. ANSWER 125°, 55° for Example 3 GUIDED PRACTICE

26. 4. Given that EFGis a right angle, find mEFH andm HFG. ANSWER 60°, 30° for Example 3 GUIDED PRACTICE

27. Trapeze The diagram 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, mDEG = m GKL. So, mGKL = 157°. ~ ~ ~ EXAMPLE 4 Identify congruent angles

28. Use the diagram shown. ANSWER T and S, P and R. for Example 4 GUIDED PRACTICE 5. Identify all pairs of congruent angles in the diagram.

29. Use the diagram shown. o o 6. In the diagram, mPQR = 130 , mQRS = 84, and m TSR = 121 . Find the other angle measures in the diagram. o m PTS = 121, m QPT = 84° ANSWER for Example 4 GUIDED PRACTICE

30. An ANGLE BISECTOR is a ray that divides an angle into two angles that are congruent. • Draw and label an acute angle, . • Fold the paper so that MN is on top of LM. • Draw a point P on the folded side. Connect it with M. • Measure the angles formed.

31. In the diagram at the right, YWbisects XYZ, and mXYW = 18 . Find m XYZ. o By the Angle Addition Postulate, m XYZ =mXYW + m WYZ. BecauseYW bisects XYZyou know thatXYW WYZ. So, m XYW = m WYZ, and you can write m XYZ = m XYW + m WYZ = 18°+ 18° = 36°. ~ EXAMPLE 5 Double an angle measure SOLUTION

32. ANSWER 90° for Example 5 GUIDED PRACTICE 7. Angle MNPis a straight angle, and NQbisects MNP. Draw MNP And NQ. Use arcs to mark the congruent angles in your diagram, and give the angle measures of these congruent angles.

33. right ANSWER o 2.m B = 62 acute ANSWER o 3.m C = 119 obtuse ANSWER Daily Homework Quiz Classify each angle as acute, obtuse , right or straight. o 1.m A = 90

34. o 4.Ifm DEG = 84 , findmFEG. o 14 ANSWER o 5. If XY bisects ZXW and m XZY = 36 , find m ZXW. o 72 ANSWER Daily Homework Quiz

35. D C B A EXAMPLE 4 Two angles that share a common vertex and side, but have no common interior points, are called ________ angles. adjacent Name the two adjacent angles in the diagram. Click to check. The common vertex is __ and the common side is ___. D

36. Classifying Angles in a Coordinate Plane • Plot the points L(-4,2), M(-1,-1), N(2,2), Q(4,-1), and P(2,-4). Then measure and classify the following angles as acute, right, obtuse, or straight. • LMN • LMP • NMQ • LMQ

37. Solution: • Begin by plotting the points. Then use a protractor to measure each angle.

38. Solution: • Begin by plotting the points. Then use a protractor to measure each angle. Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.

39. Use a protractor to draw two adjacent angles and so that is acute and is straight. Click to see a sample answer. N L O M Classify as acute, right, obtuse, or straight: Extra Example 4 obtuse

40. Draw 5 points A, B, C, D, and E so that all four statements are true: are adjacent. is obtuse. D is in the exterior of is a right angle. Click to see a sample answer. C A B D E Checkpoint Does your solution meet the requirements above?

41. QUESTIONS?