Division of Segments & Angles.

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Presentation transcript:

Division of Segments & Angles. Geometry 1.5

Definition: SEGMENT BISECTOR Is a point, segment, ray, or line that divides a segment into 2 segments. a. If a segment (or line or ray) bisects a segment, then it forms 2 congruent segments. b. If a segment (or line or ray) divides a segment into 2 congruent segments, then it bisects the segment.

Definition: SEGMENT TRISECTORS Are 2 points, lines, segments, or rays that divide a segment into 3 segments.

Definition: MIDPOINT Is a point that divides a segment into 2 segments.

Definition: ANGLE BISECTOR: Is a ray that divides an angle into 2 angles.

Definition: ANGLE TRISECTORS Are 2 rays that divide an angle into 3 angles.

Only segments have midpoints (not lines or rays) CAUTION Only segments have midpoints (not lines or rays) Questions: For a given segment, how many midpoints exist? For a given segment how many bisectors exist?

EXAMPLES

1.

Given: bisects <BAC. m<1 = 2x + 8 m<2 = x + 20 Find: a. m<1 b. m<BAC 2x + 8 = x + 20 -x - 8 = -x – 8 x = 12 2(12) + 8 = 32º= m<1 m<BAC = <1+<2 = 32 + 32 = 64º

3. Bisects <BAC = 20º20’ = 40º40’ = 49º40’ = 22º31’30” a. If m<1 = 20º20’, what is m<2? = 20º20’ b. If m<1 = 20º20’, what is m<BAC? =20º20’ + 20º20’ = 40º40’ c. If m<1 = 24º50’, what is m<BAC? =24º50’ + 24º50’ = 49º40’ d. If m<BAC = 45º3’, what is <1? =45º3’÷2 = 22º31’30”

Prove: bisects <AOD Given: Prove: bisects <AOD Statements Reasons 1. 1. Given 2. If a ray divides an angle into two congruent angles, then it bisects the angle. 2. bisects <AOD

7. Given: is divided by points B,C such that AB:BC:CD=4:1:3 7. Given: is divided by points B,C such that AB:BC:CD=4:1:3. If AD=40, find the length of AC. 8. Given: If the larger of the two numbered angles is 20 more than one half the smaller, is acute or obtuse?