Math 7 geometry 03 angles and angle measurements

CABT Math 7 Geometry Lecture 03

1. The session shall begin shortly…

2. LET’S START!

3. Angles and Angle Measurements A Mathematics 7 Presentation TXT, pp. 395-400, 406-416, 421-428, 432-435

4. An angle the union of two different rays that have the same initial point (or common endpoint). The rays are the sides of the angle. The initial point is called the vertex . Angles vertex sides A B C Angles & Angle Measurements

5. Naming Angles A B C Notation: We denote an angle with three points and “ ” symbol. Naming: Use the three points, with the vertex always in the middle . The angle can also be named with just the vertex point as long as it is a “stand alone angle”. The angle shown can be named as:  BAC ,  CAB ,  A Angles & Angle Measurements

6. Use the diagram below to answer the following questions. a. Name the vertex. R b. Name the sides of the angle. c. Name the angle in three different ways.  SRT ,  TRS ,  R R S T Check your understanding

7. Measure of an Angle A B C The MEASURE of an angle is the amount of rotation a side makes with respect to the other side. The unit is DEGREES (  ) Notation : We put the letter “m” in front of the angle name to denote its measure. The measure of the angle shown can be written as: m  BAC , m  CAB , m A Angles & Angle Measurements

8. Example : Here are some examples of angles and their degree measurements. Measure of an Angle Angles & Angle Measurements

9. Protractor A device used to measure angles. Measure of an Angle Angles & Angle Measurements

10. Measure of an Angle Angles & Angle Measurements A B C What is the measure of this angle? ANSWER: 50 

11. Measure of an Angle Angles & Angle Measurements A B C What is the measure of this angle? ANSWER: 110 

12. Measure of an Angle Angles & Angle Measurements A B C What is the measure of this angle? ANSWER: 90 

13. Measure of an Angle Angles & Angle Measurements A B C What is the measure of this angle? ANSWER: 70 

14. Reading a Protractor Date : 01/13/2014 at 15:51:52 From : Donya Ina Subject: Reading a Protractor Anak , alam kong may protractor ka at marunong kang mag-measure ng angle. At least may pakinabang ‘ yan kesa kung anu-ano ang ipino -post mo sa FB. Pero tanong ko lang : which side of a protractor scale do you use ? Paki-explain. Lab U! Angles & Angle Measurements Measure of an Angle

15. Date: 01/13/2014 at 15:52:53 From : Anak Subject: Re: Reading a Protractor Donya Ina, I think that the easiest way to decide which scale to use is to look at the angle you're measuring. If it is greater than a right angle (more than ninety degrees ) then use the scale that has numbers greater than 90. If the angle is obviously acute (measures less than 90 degrees), use the scale which has numbers less than 90. Of course you have to line up the protractor correctly with the center point on the vertex of the angle and one of the rays of the angle coincide with the line on the protractor . I-push mo lang ‘ yan ! Angles & Angle Measurements Measure of an Angle

16. Wala Lang…

17. The Angle Measurement postulate To every angle, there corresponds a unique measure r in degrees such that r is between 0 and 180. Measure of an Angle Angles & Angle Measurements

18. Measure of an Angle B A C In half-plane H 1 through the endpoint of ray BC lying in the edge of the half-plane there is exactly one ray BA, with A in H 1 , such that angle ABC formed by the two rays has a given measure between 0 and 180 degrees. Angles & Angle Measurements The Angle construction Postulate (or Protractor Postulate) WHAT?!

19. What do the two postulates mean? 1] All angles are measured between 0 and 180 . 2] They can be measured with a protractor. 3] The measurement is the absolute values of the numbers read on the protractor. 4] The values of 0 and 180 on the protractor were arbitrarily selected. Measure of an Angle Angles & Angle Measurements

20. Two angles are congruent if they have the same measure. Symbol: “ ” (is congruent to) Congruent Angles B A C Angles & Angle Measurements Y X Z  ABC and  XYZ are congruent, so we write  ABC   XYZ which means m  ABC = m  XYZ

21. Straight angle m  A = 180° Obtuse angle 90 < m  A < 180° Acute angle  < m  A < 90° Classifying Angles Angles are classified as acute , right , obtuse , and straight , according to their measures. Angles have measures greater than 0 °and less or equal to 180°. A A Right angle m  A = 90° A A Angles & Angle Measurements

22. An acute angle is an angle measuring between 0 and 90 degrees. Classifying Angles Angles & Angle Measurements

23. An obtuse angle is an angle measuring between 90 and 180 degrees.              Classifying Angles Angles & Angle Measurements

24. Classifying Angles Angles & Angle Measurements A right angle is an angle measuring exactly 90 degrees .   It is formed by two perpendicular rays with a common endpoint.             Denotes 90 degrees

25. Classifying Angles Angles & Angle Measurements A straight angle is an angle measuring exactly 180 degrees . It is formed by two opposite rays with a common endpoint.              A

26. Determine whether each of the following is acute, right, or obtuse. 1. 2. 3. 4. 5. ? ? acute obtuse right obtuse acute Check your understanding

27. Complementary Angles Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees . We say that one is the COMPLEMENT of the other. Example : These two angles are complementary.                                                                                   Angle Pairs Angles & Angle Measurements

28. These two angles can be "pasted" together to form a right angle!                           Angle Pairs Angles & Angle Measurements

29. Supplementary Angles Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. We say that one is the SUPPLEMENT of the other. Example: These two angles are supplementary.                                                                         Angle Pairs Angles & Angle Measurements

30. These two angles can be "pasted" together to form a straight line ! Angle Pairs Angles & Angle Measurements

31. 1. Find the complement of 65  . Examples. (See solution on the board) Answer : 25  Answer : 40  Angle Pairs Angles & Angle Measurements 2 . Two angles are supplementary. One measures 140  . What is the measure of the other angle? Answer : 30  3. The supplement of an angle is 30 degrees more than twice its complement. What is the angle?

32. A. Determine the complement of 1 . 48  2 . 79  42  Check your understanding 11  B. Determine the supplement of 1 . 48  2 . 123  132  57 

33. Angle Pairs Angles & Angle Measurements Adjacent Angles Adjacent angles are two angles with a common vertex and a common side but no common interior points. A B C D Adjacent angles:  BAC and  DAB

34. Angle Pairs Angles & Angle Measurements Angle Addition Postulate The sum of the measures of two adjacent angles is equal to the measure of the angle formed by the two nonadjacent rays. A B C D m  BAC + m  DAB = m  DAC

35. Angle Pairs Angles & Angle Measurements Bisector of an Angle An ANGLE BISECTOR is a ray in the interior of an angle that divides the angle into two congruent adjacent angles A B C D If ray AB is a bisector of m  DAC , then  BAC   DAB This means: m  BAC = m  DAB = ½ m  DAC

36. Angle Pairs Angles & Angle Measurements Linear Pairs Two adjacent angles form a linear pair if their uncommon sides form opposite rays B D C A  ABD and m  DBC are linear pairs

37. Angle Pairs Angles & Angle Measurements Linear Pair Postulate Linear pairs are supplementary. B D C A Since  ABD and m  DBC are linear pairs, then they are supplementary; that is, m  ABD + m  DBC = 180

38. Examples : Find x ( see solution on the board ). 2. 3. x 55  165  x x =35  x =15  Angle Pairs Angles & Angle Measurements 1. 55  x P U S H m  PUH =140 x = 85 

39. Examples : ( see solution on the board ). Angle Pairs Angles & Angle Measurements 4. P S H a. If m  HUP = 120, what is m  SUH ? U is a bisector of  HUP 60  mHUS = 74  b. If m  SUP = 74, what is m  HUS and m  PUH ? mPUH = 148 

40. Vertical Angles VERTICAL angles are two nonadjacent angles formed by two intersecting lines. Angle Pairs Angles & Angle Measurements  AED and  BEC are vertical angles  AEB and  DEC are vertical angles

41. Vertical Angle Theorem Vertical angles are congruent Angle Pairs Angles & Angle Measurements  AED and  BEC are vertical angles, so  AED   BEC or m  AED = m  BEC  AEB and  DEC are also vertical angles, so  AEB   DEC or m  AEB = m  DEC

42. Vertical Angles Note that the adjacent angles form linear pairs. Angle Pairs Angles & Angle Measurements The ff. are linear pairs:  AEB and  BEC  AED and  AEB  AED and  CED  DEC and  BEC

43. Example 1. 110  110  70  70  Angle Pairs Angles & Angle Measurements

44. Example : 2. Find the value of x , y, and z. 58  x x = 58  y = 122  z = 122  y z Angle Pairs Angles & Angle Measurements

45. 52  B A F E D C 60  G 68  68  60  52  Example 3. Find the measures of the numbered angles. 3 2 1 4 Angle Pairs Angles & Angle Measurements

46. Perpendicular Lines Lines that intersect at right angles. Angle Pairs Angles & Angle Measurements In the figure, line m is perpendicular to line n . We write: m  n m n

47. Perpendicular Bisector A line, ray, or line segment that passes through the midpoint of a segment. Angle Pairs Angles & Angle Measurements In the figure, line m is a perpendicular bisector of segment AB. It means: m is perpendicular to line AB m passes through the midpoint C, which means AC = CB m A B C D

48. 20  C J D E F G H 70  90  70  20  90  Example: Given the figure shown on the right. Find the measures of the following:  DCE  GCH  ECF  DCJ  JCH Angle Pairs Angles & Angle Measurements

49. Summing it up! Angles & Angle Measurements

50. Thank you!