1.5 Segment and Angle Bisectors

GOAL 1 Bisect a segment

GOAL 2 Bisect an Angle.

What you should learn

To solve real-life problems, such as finding the angle measures of a kite.

Why you should learn it

GOAL 1 BISECTING A SEGMENT

1.5 Segment and Angle Bisectors

Vocabulary

The point that divides a segment into two congruent segments is called the ________ of the segment. Another way to describe this point is that it _______ the segment.

midpointbisects

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.

Click to check your answers.

XZ

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

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

In geometry, a construction is a drawing that uses only two tools, a ________ and a ___________.compass straightedge

Be sure to follow the steps on page 34 to perform the construction of a segment bisector and midpoint. You must be able to complete this construction on a quiz or test, and will use it again later in the course. Click on the camera below if you want to see a video of the construction. While watching the video, you may pause at any time by clicking the pause button.

You may need to find the coordinates of the midpoint of a segment in the coordinate plane. To do this, simply find the mean (average) of the x-coordinates and the mean of the y-coordinates. In algebraic terms,

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

x1 x21 2

2

x x

y1

y2

1 2

2

y y

y

x

A(x1, y1)

B(x2, y2)

EXAMPLE 1 Did you study Example 1???

Extra Example 1

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

Click to see the solution.

DE

1 2 1 2,2 2

x x y yM

3 ( 4) 5 0,

2 2

1 5,

2 2

=

=

EXAMPLE 2

Extra Example 2The midpoint of is M(3, -4). One endpoint is Y(-3, -1). Find the coordinates of the other endpoint.

Click to see the solution.

XY

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

33

2

x and

14.

2

y

Solve: 3 6x 1 8y

9x 7y

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

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.

Click for the answers.

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.

Follow the steps on page 36 to perform the construction of an angle bisector. You must be able to complete this construction on a quiz or test, and will use it again later in the course. Click on the camera below if you want to see a video of the construction. While watching the video, you may pause at any time by clicking the pause button.

EXAMPLE 3

Extra Example 3

EXAMPLE 4

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

Extra Example 4

EXAMPLE 5

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.

Click for the solution:

wire wire

tower

47

?

2 47 94

Extra Example 5

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

Click for the solution:3 20 10x x

2 30x

15x

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

Click to check your solution:

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

QUESTIONS?