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ANGLES

ANGLES. An Angle is the union of two non-collinear rays with a common endpoint. Which of the ff. is not an angle?. a b c d e. PARTS. C interior exterior

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ANGLES

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  1. ANGLES • An Angle is the union of two non-collinear rays with a common endpoint.

  2. Which of the ff. is not an angle? a b c d e

  3. PARTS C interiorexterior A B Interior and Exterior Vertex : B Sides : BA and BC

  4. Naming Angles • 3 ways of naming ANGLES 1. Using 3 capital letters, the middle letter represents the vertex while the other must come from each of the side of the angle.

  5. Example 1 • The angle below is denoted by ABC or CBA. C B A

  6. Write all the possible names of the given angle below: A B C D

  7. Naming Angles 2. Using a lower case letter or numeral found in the interior of the vertex. The angle below is denoted by a and 1 a 1

  8. Write all the possible names of the given angle below: A B 4 C D

  9. Naming ANGLES 3. Single Vertex form, a single capital letter which represents the vertex.( NOTE: rule #3 cannot be used in adjacent Angles)

  10. Write all the possible names of the given angle below: A B 4 C D

  11. Example ( in this case you cannot name ACD as C, since there are 3 angles in it. ) A B C D

  12. REVIEW: Write the possible names for, • 1 A B • 2 5 3 4 • 3 • 4 • 5 1 2 C D

  13. Postulate 11Protractor Postulate (AngleMeasurement Postulate) • To every Angle there corresponds a real number between 0-180.

  14. Measuring Angles • PROTRACTOR – the device used in measuring angles. • DEGREE – is the unit of measuring an angle.

  15. The PROTRACTOR

  16. Kinds of angles • Acute ( between 0 – 90 ) • Right ( 90 ) • Obtuse ( between 90 -180 ) 1 2 3

  17. The PROTRACTOR • Can you addm FAD and m DAC? What will be the result?

  18. Postulate 13The Angle Addition Postulate If D is in the interior of BAC, then, m BAC = m BAD + m DAC.

  19. AAP A D B C m ABD +m DBC = ABC

  20. Let us apply AAP, E 1. What is the sum of m EDB +m BDC? D A B C Ans. m EDC

  21. Let us apply AAP, E 2.What is the sum of m ABD +m DBC? D A B C Ans. m ABC

  22. Let us apply AAP, E 3. What is the result if m EDC – m BDC? D A B C Ans. m EDB

  23. Let us apply AAP, E 4. What is the result if m CBA – m DBA? D A B C ANS. m CBD

  24. ALGEOS: Solve what is asked for. • If m CAD = 90 Find : D a) x =____ B b) m CAB =______ x 2x A C Ans. a) x = 30 b) 60

  25. ALGEOS: Solve what is asked for. 2. If m CDA = 120 Find : A a) x =____ B b) m BDA =______ 2x+10 30 D C Ans. a) 40 b) 90

  26. ALGEOS: Solve what is asked for. 3. If m BDA = m BDC Find : A a) x =____ B b) m ADC =______ 2x+60 12x D C Ans. a) 6 b) 144

  27. Kinds of angles • Acute ( between 0 – 90 ) • Right ( 90 ) • Obtuse ( between 90 -180 ) 1 2 3

  28. ANGLE PAIRS COMPLEMENTARY ANGLES SUPPLEMENTARY ANGLES

  29. Complementary Two (2) angles are complementary if the sum of their measures is exactly 90. Supplementary Two (2) angles are supplementary if the sum of their measures is exactly 180. Definitions

  30. 47 5 35 33.5 67 43 85 55 56.5 23 Find the complement of the following angles:

  31. Theorem 4-3 • Two angles which are complementaryare both acute.

  32. 1.107 2. 124 3. 47 4. 35.5 5. 171 73 56 133 144.5 9 Find the supplement of the ff. angles:

  33. There is no theorem about the supplementary angles.

  34. This is the exception to the rule, The supplementof a 90 angle is 90. These angles are both right.

  35. Theorem 4-5 • If two angles are both congruent and supplementary, then each is a right angle.

  36. Review: Translate the ff. to algebraic expressions, 1.The larger angle is four times the smaller angle. Ans.Let x = smaller angle 4x= larger angle

  37. Translate to algebraic expressions, 2. The measure of an angle is 4 times its complement. Ans.Let x = 1st angle 90 – x = the complement of the 1st angle 4( 90-x ) = new representation of the 1stangle

  38. A. Find the measure of the larger of two supplementary angles. The measure of the larger is four times the smaller. SOLUTION: Let x = smaller angle 4x = larger angle X + 4X = 180 5X = 180 X = 36 ( smaller) 4X = 144 ( larger) Solve.

  39. B. If the measure of one angle is five times its complement, find the measure of its supplement. SOLUTION: Let x = 1st angle 5x = 2nd angle X + 5x = 90 6x = 90 x = 15 ( 1st ) 5x = 75 ( 2nd) 180 – 75 = 105 The supplement is 105. Solve.

  40. 16. If the measure of an angle is three times the measure of its supplement, what is the measure of the angle? Solution: Let x = smaller angle 3x = larger angle X + 3x = 180 4x = 180 x = 45 4x = 135 The larger angle is 135. Key ( pp. 94 – 95 )

  41. 17. The measure of an angle is 24 more than the measure of its supplement. Find the measure of both angles? Solution: Let x = smaller angle x + 24= larger angle X + X + 24 = 180 2x + 24= 180 2 x = 180 - 24 2x= 156 x = 78 ( smaller) x + 24 = 102 ( larger) Key ( pp. 94 – 95 )

  42. 24. Twice the measure of an angle is 30 less than five times the measure of its supplement. What is the measure of the angle? Solution: Let x = 1st angle y= 2nd angle 1) X + Y = 180 Key ( pp. 94 – 95 )

  43. Let x = 1st angle y = 2nd angle EQ. 1) x + y = 180( since x and y are supplementary ) If the 1st angle is doubled, the 2nd angle is 30 less than 5 times the first: ( this is the new equation ) Let 2x = 1st angle ( x ) 5x – 30 = 2nd angle ( y ) Using EQ. 1 as solution ( X + Y = 180 ) 2x + ( 5x – 30 ) = 180 7x = 180 + 30 7x = 210 X = 30

  44. Using EQ. 1 ( X + Y ) = 180 Since x = 30 1st angle theny = 150 2nd angle Using the new Equation, 2x = 1st angle ( x ) 5x – 30 = 2nd angle ( y ) Then 2x = 60 1st angle 5x – 30 = 120 2nd angle

  45. Review: Name pairs of angles which are complementary and supplementary, D C E 30 30 60 60 O A B

  46. Theorem 4-7The Complement Theorem Complements of congruent angles are congruent.

  47. Theorem 4-6The Supplement Theorem • Supplements of congruent angles are congruent.

  48. 1.Which ray is common to AOEand EOC? D C E 30 30 60 60 O A B

  49. 2.Which ray is common to AODand DOB? D C E 30 30 60 60 O A B

  50. 3.Which ray is common to AODand DOC? D C E 30 30 60 60 O A B

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