Angles and Their Measures

1. Angles and Their Measures

2. Angles

3. Recap Geometrical Terms An exact location on a plane is called a point. Point A straight path on a plane, extending in both directions with no endpoints, is called a line. Line A part of a line that has two endpoints and thus has a definite length is called a line segment. Line segment A line segment extended indefinitely in one direction is called a ray. Ray

4. Angles In Daily Life If we look around us, we will see angles everywhere.

5. A B C Ray BAand rayBCare called thearmsof ABC. What Is An Angle ? When two non-collinear rays join with a common endpoint (origin) an angle is formed. Ray BA B Common endpoint Ray BC Common endpoint is called the vertex of the angle.Bis thevertexof ÐABC. Ray BA and BC are two non-collinear rays

6. Fact: We can also think of an angle formed by rotating one ray away from its initial position.

7. 1.4 Angles and Their Measures 1 2 GOAL GOAL Use Angle Postulates Classify angles as acute, right, obtuse, or straight.

8. 1.4 Angles and Their Measures 1 GOAL USING ANGLE POSTULATES C A B Naming Angles 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 Vertex Side Side

9. C A B Naming Angles The three names for this angle: The middle point is the vertex

10. Naming Angles How many angles do you see? THREE

11. Why should you not use to name any angle in the figure? 1. 2. 3. EXAMPLE Name the angles in the figure. Answers: All three angles have N as the vertex, so could mean any of the angles.

12. 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 degrees The units used to measure angles are called _______, and the symbol for them is a _.

13. Measuring With a Protractor Read OUTTER scale from LEFT to RIGHT 55º Vertex One side of the angle lined up along the LEFT half of the protractor

14. Measuring With a Protractor Read INNER scale from RIGHT to LEFT 145º One side of the angle lined up along the RIGHT half of the protractor Vertex

15. Y Let’s measure some angles. R M T Q S

16. 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:

17. For any point A on one side of , can be matched one to one with the real numbers from 0 to 180. The absolute value of the difference between the real numbers for is the ________ of Find Use either scale on the protractor to find it, but use the same one for both rays. or EXAMPLE PROTRACTOR POSTULATE measure A B Solutions: O

18. 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.

19. 1.4 Angles and Their Measures 2 GOAL CLASSIFYING ANGLES Right Acute Obtuse Straight

20. A A A A Angles are classified according to their angle measure

21. Use a protractor to draw two adjacent angles and so that is acute and is straight. N L O M Classify as acute, right, obtuse, or straight: Example obtuse

22. Right angle: An angle whose measure is 90 degrees. Straight Angle Right Angle Acute Angle Obtuse Angle

23. Examples Of Right Angle

24. Obtuse angle: An angle whose measure is greater than 90 degrees. Straight Angle Right Angle Acute Angle Obtuse Angle

25. Examples Of Obtuse Angle

26. Acute angle: An angle whose measure is less than 90 degrees. Straight Angle Right Angle Acute Angle Obtuse Angle

27. Examples Of Acute Angle

28. Straight angle: An angle whose measure is 180 degrees. Straight Angle Right Angle Acute Angle Obtuse Angle

29. Examples Of Straight Angle

30. Which of the angles below is a right angle, less than a right angle and greater than a right angle? P D R Q A E F B C 1. 2. Test Yourself 1 Greater than a right angle 3. Right angle Less than a right angle

31. Angle Addition Postulate

32. If P is in the interior of then R P S T ANGLE ADDITION POSTULATE

33. D • Name the angles in the figure. C F E • In the figure above, and Find the measure of . Checkpoint Solution:

34. Angle Addition Postulate First, let’s recall some previous information from last week…. We discussed the 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: JK + KL = JL 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… J K L

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

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

37. o ALGEBRA. Given 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 145 = (2x + 10)+ (4x – 3) o 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.

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

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

40. Given: mÐRSV = x + 5 mÐVST = 3x - 9 mÐRST = 68 Find x. Algebra Connection Slide 4 R V Extension: Now that you know x = 18, find mÐRSV and mÐVST. mÐRSV = x + 5 mÐRSV = 18 + 5 = 23 mÐVST = 3x - 9 mÐVST = 3(18) – 9 = 45 Check: mÐRSV + mÐVST = mÐRST 23+ 45 =68 S T Set up an equation using the Angle Addition Postulate. mÐRSV + mÐVST = mÐRST x + 5 + 3x – 9 = 68 4x- 4 = 68 4x = 72 x = 18 Plug in what you know. Solve.

41. mÐBQC = x – 7 mÐCQD = 2x – 1 mÐBQD = 2x + 34 Find x, mÐBQC, mÐCQD, mÐBQD. C B mÐBQC = x – 7 mÐBQC = 42 – 7 = 35 mÐCQD = 2x – 1 mÐCQD = 2(42) – 1 = 83 mÐBQD = 2x + 34 mÐBQD = 2(42) + 34 = 118 Check: mÐBQC + mÐCQD = mÐBQD 35+83 = 118 Q D mÐBQC + mÐCQD = mÐBQD x – 7 + 2x – 1 = 2x + 34 3x – 8 = 2x + 34 x – 8 = 34 x = 42 x = 42 mÐCQD = 83 Algebra Connection Slide 5 mÐBQC = 35 mÐBQD = 118

42. D 300 E F D A 300 300 E F B C Congruent Angles Two angles that have the same measure are called congruent angles. Congruent angles have the same size and shape.

43. Objectives Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles.

44. Warm Up Simplify each expression. 1.90 –(x + 20) 2. 180 – (3x – 10) Write an algebraic expression for each of the following. 3. 4 more than twice a number 4. 6 less than half a number 70 –x 190 –3x 2n + 4

45. Objectives Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles.

46. Vocabulary adjacent angles linear pair complementary angles supplementary angles vertical angles

47. Pairs Of Angles : Types • Adjacent angles • Vertically opposite angles • Complimentary angles • Supplementary angles • Linear pairs of angles

49. A D A C D Common ray B E F B C Common vertex Adjacent Angles Two angles that have acommon vertexand acommon rayare called adjacent angles. Adjacent AnglesABDand DBC ABCand DEF are not adjacent angles Adjacent angles do not overlap each other.

50. A C D B Vertically Opposite Angles Vertically opposite angles are pairs of angles formed bytwo lines intersecting at a point. ÐAPC = ÐBPD ÐAPB = ÐCPD P Four anglesare formed at the point of intersection. Vertically opposite angles arecongruent. Point of intersection ‘P’ is thecommon vertex of the four angles.