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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?