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10.2 Arcs and chords

10.2 Arcs and chords. Pg 603. Central angle. Central angle- angle whose vertex is the center of a circle. A.  ACB is a central angle. C. B. Arcs. A. Arc- a piece of a circle. Named with 2 or 3 letters Measured in degrees

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10.2 Arcs and chords

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  1. 10.2 Arcs and chords Pg 603

  2. Central angle • Central angle- angle whose vertex is the center of a circle A ACB is a central angle C B

  3. Arcs A • Arc- a piece of a circle. Named with 2 or 3 letters Measured in degrees • Minor arc- part of a circle that measures less than 180o (named by 2 letters). B B ( BP P

  4. More arcs • Major arc- part of a circle that measures between 180o and 360o. (needs three letters to name) • Semicircle- an arc whose endpts are the endpts of a diameter of the circle (OR ½ of a circle) A B ( ( ABC or CBA C C S

  5. Arc measures • Measure of a minor arc- measure of its central  • Measure of a major arc- 360o minus measure of minor arc

  6. Ex: find the arc measures ( E m AB= m BC= m AEC= m BCA= 50o ( 130o ( A 180o 180o ( D 180o+130o = 310o 50o 130o C OR 360o- 50o = 310o B

  7. Post. 26arc addition postulate • The measure of an arc formed by two adjacent arcs is the sum of the measures of those arcs. B A ( ( ( C m ABC = m AB+ m BC

  8. Congruency among arcs • Congruent arcs- 2 arcs with the same measure • MUST be from the same circle OR  circles!!!

  9. Example ( m AB=30o A ( m DC=30o E 30o B D ( ( 30o AB @ DC C

  10. Ex: continued ( m BD= 45o A ( m AE= 45o B ( ( BD @ AE The arcs are the same measure; so, why aren’t they ? 45o C D E The 2 circles are NOT  !

  11. Thm 10.4 • In the same circle (or in @ circles), 2 minor arcs are @ iff their corresponding chords are @. A ( ( AB @ BC iff AB@ BC B C

  12. Thm 10.5 • If a diameter of a circle is  to a chord, then the diameter bisects the chord and its arc. If EG is  to DF, then DC @ CF and DG @ GF ( ( E C D F G

  13. Thm 10.6 • If one chord is a  bisector of another chord then the 1st chord is a diameter. M If JK is a  bisector of ML, then JK is a diameter. K J L

  14. ( Ex: find m BC B By thm 10.4 BD @ BC. 3x+11 ( ( 3x+11=2x+47 x=36 2x+47 2(36)+47 72+47 A 119o D C

  15. Thm10.7 • In the same circle (or in  circles), 2 chords are  iff they are =dist from the center. D C DE @ CB iff AG @ AF G F A E B

  16. Ex: find CG. CF @ CG B 6 72=CF2+62 G 49=CF2+36 6 A 13=CF2 CF = ð13 C  CG = ð13 D 7 6 F 6 E

  17. Assignment

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