1.5 Notes• Segment and Angle Bisectors

The point that divides (or bisects) a segment into two congruent segments.

A segment, ray, line, or plane that intersects a segment at its midpoint

When given two points A (x1, y1) and B (x2, y2):

Ex 1) Find the midpoint of DE with endpoints D(3,5) and E(-4, 0)

2

,2

2121 yyxxM

2

05,

2

43M

2

5,2

1M

Ex 2) Find the midpoint of PQ with endpoints P(0, -7) and Q(-2, -1)

2

,2

2121 yyxxM

2

17,

2

20M

4,1 M

Ex 3) The midpoint of XY is M(3, -4). One endpoint is Y(-3,-1). Find the coordinates of the other endpoint.

2

,2

2121 yyxxM

2

1,

2

3)4,3(

yx

2

33

x

36 xx9

2

14

y

18 yy 7

7,9

Ex 4) The midpoint of JK is M(0, ½ ). One endpoint is J(2, -2). Find the coordinates of the other endpoint.

2

,2

2121 yyxxM

2

2,

2

2)2

1,0(

yx

2

20

x

20 xx 2

2

2

2

1 y

21 yy3

3,2

A ray that divides (or bisects) an angle into two adjacent angles that are congruent

BD is an angle bisector so:

Ex 5) JK bisects angle HJL. Given that HJL = 42o, what are the measures of the angles HJK and KJL?

42o

42/2 = 21o

Ex 6) A cellular phone tower bisects the angle formed by the two wires that support it. Find the measure of the angle formed by the two wires.

47(2) = 94o

Ex 7) In the diagram, MO bisects angle LMN. The measures of the two congruent angles are (3x – 20)o and (x + 10)o. Solve for x.

3x – 20 = x + 10

2x – 20 = 10

(3x – 20)o(x + 10)o

2x = 30

x = 15

Ex 8) In the diagram BD bisect angle ABC. Find x and use it to find the measures of the angles ABD, DBC, and ABC.

2x + 50 = 5x + 5

50 = 3x + 5

45 = 3x

x = 15

ABD = 2(15) + 50

= 80o

=CBD

ABC = 80(2)

ABC = 160o