Lesson 6.2 Find Arc Measures

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Presentation transcript:

Lesson 6.2 Find Arc Measures Pg 191

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 Minor arc- part of a circle that measures less than 180o (named by 2 letters). B B ( BP P

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) (need 3 letters to name) A ( ( ABC or CBA B C C S

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

Ex: find the arc measures ( 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

Arc 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

Congruency among arcs Congruent circles are two circles with the same radius Congruent arcs- 2 arcs with the same measure that are arcs of the same circle or congruent circles!!!

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

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  !

Lesson 6.3 Apply Properties of Chords Page 198

Theorem 6.5 In the same circle (or in congruent circles), 2 minor arcs are congruent if and only if their corresponding chords are congruent. A ( ( AB @ BC iff AB@ BC B C

Theorem 6.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. M If JK is a  bisector of ML, then JK is a diameter. K J L

Theorem 6.7 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

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

Theorem 6.8 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

Ex: find CG. CF @ CG B 6 72=CF2+62 G 49=CF2+36 6 A 13=CF2 CF = √13 C