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: