Download Unit 4: Parallel and Perpendicular Lines

Parallel and Perpendicular Lines
Note Book
Name:
Class Period:
Teacher’s Name:
1
Lines and Angles / Properties of Parallel Lines
D
A
C
B
H
E
G
F
Parallel Lines:
Skew Lines:
Parallel Planes:
Transversal:
t
l
m
2
Angel Pairs Formed By Transversals:
Alternate Interior Angles:
t
1
4
2
l
3
5 6
8 7
m
Same-Side Interior Angles:
t
1
4
2
l
3
5 6
8 7
m
Alternate Exterior Angles:
t
1
4
2
l
3
5 6
8 7
m
Same-Side Exterior Angles:
t
1
4
2
l
3
5 6
8 7
m
Corresponding Angles:
t
1
4
3
5 6
8 7
2
l
3
m
t
1
4
2
l
3
5 6
8 7
m
If m∠3 = 3x – 50 and m∠7 = 2x, what is the value of x?
If m∠1 = 2x + 50 and m∠8 = 3x - 20, what is the value of x?
Find the value of each of the variables.
a° b°
d° c°
h°
4
e° 120°
g° f °
Proving Lines Parallel
Converse of the Alternate Interior Angles Theorem: If two lines
and a transversal form Alternate Interior angles that are congruent, then
the lines are parallel.
Converse of the Same-Side Interior Angles Theorem: If two
lines and a transversal form Same-Side Interior angles that are
supplementary, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem: If two lines
and a transversal form Alternate Exterior angles that are congruent, then
the lines are parallel.
Converse of the Same-Side Exterior Angles Theorem: If two
lines and a transversal form Same-Side Exterior angles that are
supplementary, then the lines are parallel.
Converse of the Corresponding Angles Theorem: If two lines and
a transversal form corresponding angles that are congruent, then the lines
are parallel.
5
t
n
1 5
2 6
3 7
4 8
9
13
10 14
11 15
12 16
l
m
Use the given information to determine which lines, if any, are
parallel. Justify each conclusion with a theorem or postulate.
 ∠14 is supplementary to ∠15
6

∠4 is supplementary to ∠16

∠4 ≅ ∠2

∠1 ≅ ∠6

∠9 ≅ ∠2

∠7 ≅ ∠12

∠4 ≅ ∠13

∠1 ≅ ∠11
Parallel and Perpendicular Lines
In a plane, if two lines are parallel to the same line, then they are
parallel to each other.
In a plane, if two lines are perpendicular to the same line, then they
are parallel to each other.
In a plane, if a line is perpendicular to one of two parallel lines,
then it is also perpendicular to the other.
7
Parallel Lines and Triangles
Triangle Sum Theorem: The sum of the measures of the angles of a
triangle is 180.
Exterior Angle of a Polygon: An angle formed by a side and an
extension of an adjacent side.
Remote Interior Angles: The two nonadjacent interior angles
corresponding to each exterior angle of a triangle.
Triangle Exterior Angle Theorem: The measure of each exterior
angle of a triangle equals the sum of the measures of its two remote
interior angles.
Find the value of each variable.
32°
75°
8
42°
x° y°
z°
Equations of Lines in the Coordinate Plane
Slope:
Slope-Intercept Form:
Point-Slope Form:
Find an equation if you are given the slope and a point:
Find an equation if you are given two points:
9
Slopes of Parallel and Perpendicular Lines
Slopes of Parallel Lines
 If two nonvertical lines are parallel, then their slopes are
equal.
 If the slopes of two distinct nonvertical lines are equal, then
the lines are parallel.
 Any two vertical lines or any two horizontal lines are
parallel.
Slopes of Perpendicular Lines
 If two nonvertical lines are perpendicular, then their slopes
opposite reciprocals.
 If the slopes of two lines are opposite reciprocals of each
other, then the lines are perpendicular.
 Any horizontal line and vertical line are perpendicular.
1
Write the equation of the line that is parallel to y = x – 3
2
and goes through the point (6 , -3).
Write the equation of the line that is perpendicular to
1
y = x – 3 and goes through the point (6 , -3).
2
10
Constructing Parallel and Perpendicular Lines
Constructing Parallel Lines
Given: line l and point N not on l
Construct: line m through N with m ǁ l
N
l
Step 1: Label two points H and J on l. Draw ⃡HN.
Step 2: At N, construct ∠1 congruent to ∠NHJ. Label the
new line m.
mǁl
11
Perpendicular at a Point on a line
Given: point P on line l
Construct: ⃡CP with ⃡CP | l
l
P
Step 1: Construct two points on l that are equidistant from
P. Label the points A and B.
Step 2: Open the compass wider so the opening is
1
greater than AB. With the compass tip on
2
A, draw an arc above point P.
Step 3: Without changing the compass setting, place the
compass point on point B. Draw an arc that
intersects the arc from step 2. Label the point of
intersection C.
⃡ .
Step 4: Draw CP
⃡𝐂𝐏 | l
12
Perpendicular from a Point to a Line
Given: line l and point R not on l
Construct: ⃡RG with ⃡RG | l
R
l
Step 1: Open your compass to size greater than the
distance form R to l. With the compass on point R,
draw an arc that intersects l at two points. Label the
points E and F.
Step 2: Place the compass point on E and make an arc.
Step 3: Keep the same compass setting. With the compass
tip on F, draw an arc that intersects the arc from
step 2. Label the point of intersection G.
Step 4: Draw ⃡RG.
⃡𝐑𝐆 | l
13