Download Theorem 4.8 By - Coweta County Schools

4.7
Theorems On Perpendicular Lines
Theorem 4.7:
If two lines intersect to form a
linear pair of congruent
angles, then the lines are
______________.
perpendicu lar
If 1  2, then g ____
 h.
g
1
2
h
4.7
Theorems On Perpendicular Lines
Theorem 4.8:
If two lines are perpendicular,
then they intersect to form
four ______________.
right angles
If a  b, then 1, 2, 3, and 4
and 4 are __________
____.
right angles
b
1
2
3
4
a
4.7
Theorems On Perpendicular Lines
Theorem 4.9:
If two sides of two adjacent
acute angles are
perpendicular, then the
angles are ________________.
complement ary
If BA  BC, then 1 and 2
are __________
complement____.
ary
A
1
2
B
C
4.7
Theorems On Perpendicular Lines
Theorem 4.10: Perpendicular Transversal Theorem
If a transversal is perpendicular
to one of two parallel lines, then
it is _____________
perpendicu lar to the other.
If h k and j  h, then j ___
 k.
j
h
k
4.7
Theorems On Perpendicular Lines
Theorem 4.11: Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are
perpendicular to the same line,
then they are _________
parallel to each
other.
If m  p and n  p, then m ___ n.
m
n
p
4.7
Theorems On Perpendicular Lines
g
2
K
1
2
h
N
J
1
13
L
13
2
M
4.7
Theorems On Perpendicular Lines
Example 1 Application of the Theorems
Find the value of x.
a.
a
b
l
b.
xo
xo
55o
p
Solution
perpendicular all four angles
a. Because a and b are _______________,
formed are right angles by _______________.
Theorem 4.8
90
By definition of a right angle, x = ____.
b. Because l and p are perpendicular, all four angles
Theorem 4.8
formed are right angles by ____________.
Theorem 4.9 the 55o angle and the xo angle are
By ___________,
______________.
complementary Thus x + 55 = 90, so x = ____.
35
4.7
Theorems On Perpendicular Lines
Checkpoint. Find the value of x.
1.
2 x  x  90
3x  90
x  30
o
2x
o
x
o
o
By Theorem 4.8, all angles are right angles.
o
4.7
Theorems On Perpendicular Lines
Checkpoint. Find the value of x.
2.
3x  x  90
4 x  90
x  22.5
o
xo
3x o
o
By Theorem 4.8, all angles are right angles.
o
4.7
Theorems On Perpendicular Lines
Example 2 Find the distance between a point and a line
What is the distance from point B to
line q?
Solution
You need to find the slope of line q.
Using the points (3, 2) and (6, 5), the
slope of line q is
5 2
1 .
m
 ___
6 3
B 3, 8
q
6, 5
3, 2
1
1
The distance from point B to line q is the length of the
perpendicular segment from point B to line q. The slope of the
perpendicular segment from point B to line q is the negative
1

reciprocal of ___,
1
1 or _____ = ____.
1
4.7
Theorems On Perpendicular Lines
Example 2 Find the distance between a point and a line
What is the distance from point B to
line q?
Solution
The segment from (6, 5) to (3, 8) has
a slope of ____.
 1 So, the segment is
perpendicular to line q.
Find the distance between (6, 5)
and (3, 8).
d
3   __
6  __
5  __
8
__
2
2
B 3, 8
q
6, 5
3, 2
1
1
4.2
 ____
The distance from point B to line q is about ____
4.2 units.
4.7
Theorems On Perpendicular Lines
5.2
Use Perpendicular Bisectors
Theorem 5.3: Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant
from the endpoints of a
segment, then it is on the A
____________
perpendicu
lar _________
bisector
of the segment.
C
P
B
D
If DA  DB, then D lies on the 
________
bisector of AB.
5.2
Use Perpendicular Bisectors
Example 1 Use the Perpendicular Bisector Theorem
AC is the perpendicular
bisector of BD. Find AD.
Solution
7 x7x– 66
B
C
A
AB
AD  ____
D
___  ______
6
___
3x  ___
x  ___
2
42  ___
4 x  ____
8 .
AD  ____
44xx
5.2
Use Perpendicular Bisectors
Example 2 Use perpendicular bisectors
In the diagram, KN is the
perpendicular bisector of JL.
a. What segment lengths in
the diagram are equal?
b. Is M on KN?
Solution
K
N
J
13
L
13
a. KN bisects JL, so ____
____.
NJ = NL
Because K is on the perpendicular bisector
M
of JL, ____
KJ = ____
KL by the Theorem 5.2.
The diagram shows that ____
MJ = ____
ML= 13.
equidistant from J and L.
b. Because MJ = ML, M is ____________
So, by the Converse
__________________________________________,
of the Perpendicular Bisector Theorem
M is on the perpendicular bisector of JL, which is KN.
5.2
Use Perpendicular Bisectors
Checkpoint. In the diagram, JK is the
perpendicular bisector of GH.
F
1. What segment lengths are equal?
KG  KH
JG  JH
FG FH
2. Find GH.

x 1
4.1
4.1
J
G
K
2x x 1
2x
x+1
GH  2 x  x  1 21  1  1 4
H
5.2
Use Perpendicular Bisectors
Theorem 5.4: Concurrency of Perpendicular Bisector of a Triangle
B
The perpendicular bisector of a
triangle intersects at a point that
is equidistant from the vertices of
the triangle.
D
A
If PD, PE, and PF are perpendicular
bisectors, then PA = _____
PC
PB = _____.
P
F
E
C
5.2
Use Perpendicular Bisectors
Example 3 Use the concurrency of perpendicular bisectors
N
The perpendicular bisectors of
MNO meet at point S. Find SN.
7
Q
P
S
9
Solution
2
R
M
Using ______________,
you know that point S is
Theorem 5.4
_____________
equidistant from the vertices of the triangle.
So, _____
SO
SN = _____.
SM = _____
_____
SM
SN = _____
9
_____
SN = _____
Theorem 5.4.
Substitute.
O
5.2
Use Perpendicular Bisectors
Checkpoint. Complete the following exercise.
3. The perpendicular bisector of
ABC meet at point G. Find GC.
B
12
By ______________,
Theorem 5.4
D
_____
GC
GB = _____.
GA = _____
GC  GA
GC  15
E
G
15
A
6
F
C
5.2
Use Perpendicular Bisectors
Pg. 284, 5.2 #1-10