(1.5) Division of Segments and Angles!!!

By: Lauren Coggins, Kanak Chattopadhyay, and Morgan Muller

What?!

Definitions and Their Converses

ODefinitions are ALWAYS Reversible.

OTheorems are NOT ALWAYS Reversible. (Their converses are not always true.)

MidpointsDefinition: If a point is the midpoint of a segment, then it divides the segment into two segments.

Ex. Given: C is the midpoint of AB Conclusion: AC CB A

C

B

1.) C is the midpoint 1.) GOf AB

2.) AC CB 2.) If a point is the midpoint of a segment, then it ÷s the segment into 2 segments

MidpointsConverse: If a point divides a segment into two segment, then it is the midpoint of the segment.

Ex. Given: PI IE Conclusion: I is the midpoint of PE

PI

E

1.) PI IE 1.) G

2.) I is the midpoint of PE 2.) If a point ÷s the segment into 2 segments, then it is the midpoint of the segment

Sample Problems O If M is the midpoint of FE, what conclusions can we draw?O Conclusions:O -AM MB (If a point is the midpoint of a segment, then it

divides the segment into 2 congruent segments.)O Point M bisects AB

A

M

B

Trisection PointsDefinition: If two points trisect a segment, then they divide the segment into three congruent segments.

Ex. Given: A and K are trisection points of CE.

Conclusion: CA AK KE.

C A K E

S R

1.) A and E are 1.) G trisection points.

2.) CA AK KE 2.) If 2 points ÷ a segment into 2 segments, then they trisect the segment

Trisection Points

Converse: If two points divide a segment into three congruent segments, then they trisect the segment.

Ex. Given: CA AN NE.

Conclusion: A and N trisect CE.C A N E

1.) CA AN NE 1.) G

2.) A and N trisect CE 2.) If 2 points ÷ a segment into 3 segments, then it is the midpoint of the segment

Angle BisectorDefinition: If a ray bisects an angle, then it divides the angle into 2 angles.

Ex. Given: IN bisects MIT Prove: MIN TIN

I

T

MN

1.) IN bisects MIT 1.) G

2.) MIN TIN 2.) If a ray bisects an angle, then it ÷s the angle into 2 angles

Angle Bisector

Converse: If a ray ÷s an into 2 s, then it bisects the

Ex. Given: MIN TIN Conclusion: IN bisects MIT

1.) MIN TIN 1.) G

2.) IN bisects 2.) If a ray ÷s an into 2 s,MIT then it bisects the

I

T

MN

Sample Problems

O If OB is the bisector of AOC, then AOB is congruent to COB.

• (If a ray bisects an angle then it divides the angle into 2 congruent angles).

O

A

B

C

Angle Trisectors…

Definition: Two rays that divide an angle into three congruent angles trisect the angle. The two dividing rays are called trisectors of the angle. Definition in “if then” Form: If 2 rays trisect an angle, then they divide the angle into three congruent angles.

A

B T

H

S

Converse: If 2 rays divide the angle into 3 congruent angles, then they trisect the angle.

For Example…

If BAT TAH HAS, then AT and AH trisect BAS.

Converse: If AT and AH trisect BAS, then BAT TAH HAS.

P

I

EN

S

Sample ProblemsGiven: PS SEConclusion: S is the midpoint of PE

Reason: If a point ÷s a segment into 2 segments, then it is the midpoint of the segment.

Sample Problems

M

I L K S

Given: Points L and K are trisectors of ISConclusion: IL LK KS

Reason: If 2 points trisect a segment, then they divide the segment into 3 segments.

Sample Problems

S

N

A P

Given: ASN PSAConclusion: SA bisects PSN

Reason: If a ray divides an angle into 2 angles, then it bisects the angle.

QUIZ TIME!!!!!!!Bisector Problems…

Find CAR if AR bisects CAE and CAE equals

1.) 8040

2.) 74 1837 9

3.) 54 2227 11

4.) 30 ½15 15

5.) 26 3813 19

C

A

R

E

QUIZ TIME!!!!!!!

Given: LK bisects TI and RE TR = 6x; IE = 8x TL = 9; RK = 5 Perimeter of TREI = 84Find: IE

T I

ER

L

K

Answer = x = 4 IE = 32 units

QUIZ TIME!!!!!!!

S O L

I

D

5x

7x

3x

OD and OI divide straight angle SOL into three angles whose measures are in the ratio 5:7:2. Find mDOI.

Answer: x = 12 mDOI = 84

Works Cited

Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New

York: McDougal, Little & Company, 1991. Print.

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