Download Parallel and Perpendicular Lines

Geometry (Holt 3-4)
K.Santos
Perpendicular Bisector
Perpendicular bisector of a segment is a line
perpendicular to a segment at the segment’s midpoint.
(could be a segment or ray)
s
M
t
Line s is perpendicular to line t at it’s midpoint M
Distance from a point to a line
The shortest segment from a point to a line is
perpendicular to the line.
Distance form a point to a line is the length of the
perpendicular segment from the point to the line.
Theorem (3-4-1)
If two intersecting lines form a linear pair of congruent
angles, then the lines are perpendicular.
Given: <1 and < 2 are a linear pair
Then: m ⊥ n
n
12
m
Perpendicular Transversal Theorem
(3-4-2)
In a plane, if a line is perpendicular to one of two
parallel lines, then it is perpendicular to the other line.
a
Given: a||b
t ┴a
Then: t ┴b
b
t
Theorem (3-4-3)
Theorem:
If two coplanar lines are perpendicular to the
same line, then the two lines are parallel to each other.
Given: a ┴ t
b ┴t
Then: a||b
a
b
t
Proof of previous theorem
Given: a ┴ t
b ┴t
Prove: a||b
1
a
2
b
t
Statements
1. a ┴ t, b ┴t
2. < 1 and < 2 are right angles
3. <1 ≅<2
4. a||b
Reasons
1. Given
2. Definition of perpendicular lines
3. all right angles are congruent
4. If corresponding angles are
congruent then the lines are parallel
Theorem
If two coplanar lines are parallel to the same line,
then they are parallel to each other.
t
Given: a ||b
b ||c
Then: a || c
a
b
c
Example:
Use the picture at the right to answer the questions below:
C
P
B
x–8
12
Name the shortest segment from point A to 𝐵𝐶.
𝐴𝑃
Write and solve an inequality for x.
AC > AP
x – 8 > 12
x > 20
Example
Given the information below what can you conclude about lines a
and d?
a || b
b┴c
c || d
a ___ d?
a
b
c
Draw a picture with all the line
in it and then make a conclusion
about lines a and d.
a┴d
d
Proof
Given: r||s
<1 ≅ <2
Prove: r ⊥ s
Statements
1. r||s
2. <2 ≅ <3
3. <1 ≅ <2
4. <1 ≅ <3
5. r ⊥ s
t
1 3
r
2
s
Reasons
1. given
2. Corresponding Angles postulate
3. given
4. Transitive Property (2, 3)
5. If two intersecting lines form a
then the lines are perpendicular