Download Lesson 6 Parallel and Perpendicular lines

Lesson 6
Parallel and Perpendicular lines
There exist different types of lines that will be explored in Geometry.
Parallel Lines are coplanar lines (lines that lie on the same plane) that do not intersect.
The symbol ____
ll means “is parallel to.”
Arrows/triangles
indicate that the lines
above are parallel
So, p ll q
Perpendicular Lines are two lines that intersect to form right angles. The
 means “is perpendicular to.”
symbol ___
Two lines are perpendicular lines if they intersect to form 4 right angles. Lines
t and u are perpendicular lines.
So, t

u
Skew Lines are two non-coplanar lines; skew
lines are not parallel and do not intersect.
•Not on the same plane
•Not parallel
•Not perpendicular
•Do not intersect
Line m lies on plane A.
Line l passes through plane A.
So, line m and line l are skew.
Ex. 1
Determine whether the lines are parallel, perpendicular, or neither.
a. line v and line x
neither
b. line v and line y
parallel
c. line w and line x
perpendicular
Ex. 2
Determine whether the lines are skew. Explain your reasoning.
a. line k and line m
They are not skew because the lines intersect.
b. line l and line m
They are not skew because the lines are on the same plane
(coplanar).
Theorems about Perpendicular Lines
There exists four theorems about perpendicular lines.
Theorem 1: All right angles are congruent.
Theorem 2: If two lines are perpendicular, then they intersect to
form four right angles.
1 = 90o
2 = 90o
3 = 90o
4 = 90o
*Even if only one angle is marked as a right
angle, the other 3 are automatically right
angles.*
Ex. 4 In the diagram, HJ HL and mLHK = 42o.
Find the value of z.
Open your textbooks
•
•
•
•
P 110 #2-5, #10-14 (even)
P 111 #15-19 (odd)
P 113 #42-43
P 118 # 18-24 (even)
10
A transversal is a line that intersects two or more lines,
each at a different point. When a transversal intersects
two or more lines at different points, it forms eight angles.
t
1 2
4
3
m
6
5
7
p
8
transversal
11
Two angles are corresponding angles if they have
matching positions. Corresponding angles also lie on the
same side of a transversal.
t
1 2
To find the
corresponding
angles , draw a
“F” using the
transversal.
4
3
5
7
6
8
r
s
Corresponding Angles:
1 and 5
3 and 7
2 and 6
4 and 8
12
Corresponding Angles Theorem: If two parallel lines
are cut by a transversal, then the pairs of
corresponding angles are CONGRUENT.
120◦
120◦
r
s
13
Two angles are alternate interior angles of they lie on
opposite sides and at opposite ends of a transversal.
t
1 2
To find the
alternate interior
angles , draw a
“Z” using the
transversal.
Alternate interior
means “in
between” the
lines.
4
3
5
7
r
s
6
8
Alternate Interior Angles:
3 and 6
4 and 5
14
Unit 2 - Parallel
and Perpendicular
Lines
Alternate Interior Angles Theorem: If two parallel
lines are cut by a transversal, then the pairs of
alternate interior angles are CONGRUENT
r
70◦
70◦
s
15
Two angles are alternate exterior angles of they lie on
opposite sides and at opposite ends of a transversal.
t
To find the
alternate exterior
angles , draw a “Z”
using the
transversal.
Alternate exterior
means “outside”
the lines.
1 2
4
3
5
7
6
8
r
s
Alternate Exterior Angles:
1 and 8
2 and 7
16
Alternate Exterior Angles Theorem: If two
parallel lines are cut by a transversal, then the pairs
of alternate exterior angles are CONGRUENT.
130◦
r
s
130◦
17
• Two angles are same-side interior angles if they lie between the
two lines on the same side of the transversal.
t
To find the same –
side interior angles
, draw a “C” using
the transversal.
1 2
4
3
5
7
r
s
6
8
Same Side Interior Angles:
3 and 5
4 and 6
18
Same Side Interior Angles Theorem: If two parallel
lines are cut by a transversal, then the pairs of same –
side interior angles are SUPPLEMENTARY
r
70◦
110◦
s
19
Ex. 1 Identify the relationship between the angles.
1 5
3
a) 1 and 2
same – side interior angles
b) 4 and 6
corresponding angles
c) 3 and 6
alternate exterior angles
2 4
6
20
Ex. 2 List all the pairs of angles that fit the description.
1
2
3
6
8
4
7
5
a) Corresponding
b) Alternate Interior
c) Alternate Exterior
d) Same – Side Interior
Class Work
Pg 123 #2-14 (even)
Pg 124 #16,18
Pg 125 #34 - 35, 40- 41, 47- 48
Pg. 132 #16-18
Pg. 133 #20, 26-28, 32– 34 (even)
Pg. 134 36