Segment and AngleBisectors

1.5 Segment and Angle Bisectors

GOAL 1 Bisect a segment

GOAL 2 Bisect an Angle.

GOAL 1 BISECTING A SEGMENT

1.5 Segment and Angle Bisectors

Vocabulary

The point that divides, or bisects, a segment into two congruent segments is called the ________ of the segment.

midpoint

ZX Y

In the diagram, Y is the midpoint of if Y is on and This is shown by the red congruence marks.

XZ XZ.XY YZ

A segment, ray, line, or plane that intersects a segment at its midpoint is called a _______________.segment bisector

ZX Y

T

U

Name all eight different bisectors of shown above. XZ

, , , , , , ,UT UT YU YT UT UY TU TY YU YT�������������������������������������������������������������������������������������������� �������

All eight figures named above are bisectors of .XZ�������������� �

x1 x21 2

2

x x

y1

y2

1 2

2

y y

y

x

A(x1, y1)

B(x2, y2)

THE MIDPOINT FORMULA

If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then

the midpoint of has coordinatesAB

1 2 1 2

2 2,

x yx y

Example: Find the midpoint

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

1 2 1 2,2 2

x x y yM

3 ( 4) 5 0,

2 2

1 5,

2 2

=

(–.5, 2.5)

Solution

(x1, y1) (x2, y2)

=

=

Example: Find the endpoint

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

XY

If the coordinates of X are (x, y), then by the Midpoint Formula

Solve: 3 6x 1 8y

9x 7y

So the other endpoint is X(9, -7).

1 2 1 2

2 2,

x yx y

and

Checkpoint1. Find the midpoint of with endpoints P(0, -7) and

Q(-2, -1).

2. The midpoint of is One endpoint is J(2, -

2). Find the coordinates of the other endpoint.

PQ

JK1

0, .2

M

( 1, 4)M

( 2,3)K

GOAL 2 BISECTING AN ANGLE

1.5 Angles and Their Measures

Vocabulary

A ray that divides an angle into two adjacent angles that are congruent is called an ____________.angle bisector

F

H

G

K

In the diagram, bisects since

HK��������������

FHG.FHK GHK

m FHK m GHK Note the red congruence marks.

Example bisects Given that what are the measures of Hint: Draw a sketch!

Click to see a sample sketch.

JK��������������

.HJL 42 ,m HJL and ?HJK KJL

H

J

L

K42°

Click for the answer.

42 2m HJK m KJL

21

ExampleA 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.

Solution:

wire wire

tower

47

?

2 47 94

Example

In the diagram bisects The measures of the two congruent angles are Solve for x.

MO��������������

.LMN(3 20) and ( 10) .x x

M

L O N

(3 20)x ( 10)x

3 20 10x x 2 30x

15x

Solution:

CheckpointBD��������������

.ABCIn the diagram bisects Find x and use it to find , , and .m ABD m DBC m ABC

A

D

CB

(2 50)x (5 5)x

2 50 5 5x x 45 3x15 x

(2 50)m ABD x (2 15 50) 80

(5 50)m DBC x (5 15 5) 80

2 80m ABC 160

Solution: