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ANGLES

ANGLES. DEF of an Angle: An angle is FORMED BY TWO RAYS WITH A COMMON ENDPOINT . The Rays are the sides of the angle. The endpoints of the rays are the vertex. side. side. NAMING ANGLES. EXTERIOR POINTS. E. The MIDDLE LETTER OF the NAME ALWAYS VERTEX. USE 3 LETTERS

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ANGLES

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  1. ANGLES • DEF of an Angle: An angle is FORMED BY TWO RAYS WITH A COMMON ENDPOINT. • The Rays are the sides of the angle. • The endpoints of the rays are the vertex

  2. side side NAMING ANGLES EXTERIOR POINTS E The MIDDLE LETTER OF the NAME ALWAYS VERTEX • USE 3 LETTERS • ONE FOR THE VERTEX • ONE FOR A POINT ON EACH SIDE OF THE ANGLE J A INTERIOR POINTS vertex ABC B C

  3. ALTERNATIVE ANGLE NAMES USE1 LETTER (THE VERTEX) OR 1 NUMBER(THE VERTEX) FOR THE NAME. WE SAY side side A 4 vertex B C 4 or B

  4. Name the angles in this the diagram… T M E V 1 2 • How does your answer change if I ask for the distinct angles in the diagram?

  5. k 30° ANGLE NOTATION - ANGLES ARE MEASURED IN DEGREES. - WHEN WE WANT TO STATE THE MEASURE OF AN ANGLE WE USE LOWER CASE m IN FRONT OF THE NAME. Ex. WE SAY mk = 30°

  6. If two angles have the same measure, we say ABC  CBJ CONGRUENT ANGLES REFERS TO ANGLES WITH THE SAME MEASURE. A 20° C B 20° J

  7. C M T 3 K 5 P L NAME THE ANGLES • If the angles both measure 65° then write a statement using proper notation.

  8. CLASSIFYING ANGLES 25° Angles whose measures are less than 90° are called _________________? ACUTE ANGLES Angles whose measures are 90° are called _________________? RIGHT ANGLES 115° Angles whose measures are greater than 90° but less than 180° are called _________________? OBTUSE ANGLES

  9. C K P and Angles whose measures equal 180° are called _________________? STRAIGHT ANGLES A straight angle is a line, Formed by two opposite rays. Name the opposite rays that form the above angle…

  10. Think about this situation… V P T C m TPV = 110° m TPC = 60 ° Find m CPV = 50° This leads to the next postulate…

  11. P Q R S ANGLE ADDITION POSTULATE (1.8) • IF R IS in THE INTERIOR OF PQS, THEN m PQR + m RQS = m PQS

  12. EXAMPLES C G F D m CDF = 115° m CDG = 3x + 5 m GDF = 2x X = 22°

  13. EXAMPLES B A C D O m AOB = 4X – 2 m BOC = 5X + 10 m COD = 2X + 14 Find m AOD = 110°

  14. ANGLE BISECTOR • A Ray or Segment that DIVIDES AN ANGLE INTO CONGRUENT ANGLES • CONGRUENT ANGLES HAVE THE SAME MEASURE A B D ABD  DBC C IS AN ANGLE BISECTOR OFABC

  15. EXAMPLES A B C D IF IS AN ANGLE BISECTOR OF ACD, FIND m ACB. = 45°

  16. EXAMPLES F E G T IF IS AN ANGLE BISECTOR OF EFG, and mEFT = 4x + 12 and mTFG = 6x Find x: FIND m EFG. X = 6 m EFG = 72

  17. EXAMPLES H G K J IF IS AN ANGLE BISECTOR OF GHK, and mGHJ = 6x + 10 and mJHK = 3x + 28 Find x: FIND m GHK.

  18. B A C D O ANGLE RELATIONSHIPS ADJACENT ANGLES: ANGLES THAT SHARE A COMMON SIDE and VERTEX, BUT NO COMMON INTERIOR POINTS Name pairs of adjacent angles

  19. EXAMPLES G C D F Are thes angles adjacent angles? m GDF = 65° Find m CDG = 115°

  20. ANGLE RELATIONSHIPS • LINEAR PAIR: ADJACENT ANGLES WHOSE NON-COMMON SIDES ARE OPPOSITE RAYS. That is the two angles form a straight angle. 1 2

  21. ANGLE RELATIONSHIPS • Make observations about the angles in the worksheet. • THE SUM OF THE MEASURES OF THE ANGLES IN A LINEAR PAIR IS 180

  22. EXAMPLES G C D F Are thes angles a linear pair? m GDF = 7x + 2 and m CDG = 3x + 8 Find x and each angle. X = 17 m GDF = 121° m CDG = 59°

  23. EXAMPLE The sum of the measures of the angles in a linear pair is 180° C B 2X - 10 Y 60° Z K FIND x FIND Y FIND Z H

  24. 2 4 3 5 ANGLE RELATIONSHIPS • VERTICAL ANGLES: TWO NONADJACENT ANGLES FORMED BY TWO INTERSECTING LINES. Name a pair of vertical angles

  25. ANGLE RELATIONSHIPS Vertical angles are congruent C B 100° 4Y 80° A 2X K H FIND MKAH FIND MKAB FIND X Find Y 2x = 100 x = 50 2y = 80 y = 20

  26. EXAMPLES C B 10x + 5 2x + 21 I H K m BIK = 10X + 5 m HIC = 2X + 21 Find X Find m BIK Find m BIC

  27. ANGLE RELATIONSHIPS • PERPENDICULAR LINES: INTERSECTING LINES THAT FORM 4 RIGHT ANGLES

  28. a b d c

  29. ANGLE RELATIONSHIPS COMPLEMENTARY ANGLES…TWO ANGLES WHOSE MEASURES ADD UP TO 90° SUPPLEMENTARY ANGLES…TWO ANGLES WHOSE MEASURES ADD UP TO 180°

  30. EXAMPLES Two angles are complementary and one angle has 3 times the measure of the other. Find the two angles. Hint write an equation and solve. x + 3x = 90 4x = 90 X = 22.50 So one angle is 22.5° and its complement is 90 ° - 22.5° = 67.50 °

  31. EXAMPLES Two angles are supplementary and one angle measures 40 less than three times the other. Find both angles. Hint write an equation and solve. x + 3x - 40 = 180 4x = 220 X = 55 So one angle is 55° and its supplement is 180 ° - 55° = 125°

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