Download 1 Parallel lines and Transversals

3-1 Parallel lines and Transversals
A. Lines
1. __________________ ___________ -Two lines that are coplanar and never
meet or cross.
we say PQ||RS
P
R
Q
S
2. __________________ ___________ -Two lines that do not intersect yet are
not parallel because they are not in the same plane.
Ex 1: Name all segments that are skew to TK
3. A line that intersects two or more ___________ in a plane at different points is
called a _______________.
B. Angles
1. When a transversal intersects two lines, ___________ angles are formed.
a.) exterior angles ___________________________
b.) interior angles ___________________________
c.) consecutive interior angles ___________________________
d.) alternate exterior angles ___________________________
e.) alternate interior angles ___________________________
f.) corresponding angles ___________________________
1
2
3
5
7
4
6
8
Ex 2: Identify the special angle pair name, and tell what line the transversal is.
a.) <1 and <9 ______________ b.) <13 and <12 _______________
c.) <5 and <16 _____________ d.) <4 and <11_______________
Geometry 3-2 Angles and Parallel Lines
A. Parallel lines and angle pairs
1. Corresponding Angles Postulate -If 2 parallel lines are cut by a transversal,
then each pair of corresponding angles is congruent.
<1 <5, <2 <6, <3 <7, <4  <8
1
2
3
4
l
5
6
7
8
m
Example 1: If l || m, and m<1 = 142, find m<8
2. Theorem 3-1 -___________________ ________________ ____________
__________________ -If two parallel lines are cut by a transversal, then each
pair of alternate interior angles is congruent.
So m<_______ m<________ and mm
3. Theorem 3-2 -Consecutive Interior Angles Theorem -If two parallel lines are
cut by a transversal, then each pair of consecutive interior angles is
supplementary.
So m<_______ m<________ = ___________ and
mm
4. Theorem 3-3 -Alternate Exterior Angles Theorem -If two parallel lines are cut
by a transversal, then each pair of alternate exterior angles is congruent.
So m<_______ m<________ and mm
5. Theorem 3-4 -Perpendicular Transversal Theorem -In a plane, if a line is
perpendicular to one of two parallel lines, then it is perpendicular to the other.
Example 2: p q and t is a transversal,
p
a.) Find m<1 if m<3 = 50
b.) Find m<4 if m<5 = 145
c.) Find m<2 if m<7 = 30
d.) Find m<8 if m<2 = 40
1
2
q
3
4
5
6
7
8
Example 3: In the figure, MA || HT and NG || EL , find the values of x, y, z, and
the measure of <NMR .
N
R
E
M
A
2x
4z
O
H
T
(5y +2)o
G
L
72
C
Geometry 3-3 Slopes of lines
A. In a coordinate plane, the slope of a line is the ratio of its vertical rise to its
horizontal run.
1. Definition of Slope -the slope of a line containing two points with coordinates
(x1, y1) and (x2, y2) is given by the formula:
m = (y1 – y2), where x1 ≠ x2
(x1 – x2)
Example 1: You have points (0, 2) and (7, 3), find m.
a. If a line is horizontal, its slope is 0.
b. If a line is vertical, its slope is undefined.
Example 2: For one manufacturer of camping equipment, between 1990 and
2000 annual sales increased by 7.4 million per year. In 2000, the total sales were
85.9 million. If sales increase at the same rate, what will be the total sales in
2010?
2. Postulate 3-2 -Two non-vertical lines have the same slope if and only if they
are parallel.
3. Postulate 3-3 -Two non-vertical lines are, iff the product of their slopes is –1.
Example 3: Given A(-3,-2), B(9,1), C(3,6), and D(5,-2). Determine if AB is
parallel or perpendicular to CD.
Example 4: Graph the line that contains Q(5, 1) and is parallel to MN with M (-2,
4) and N (2, 1).
Geometry 3-4 Equations of Lines
A. Forms of Equations
1. Slope-Intercept Form – The slope-intercept form of the equation of a line is y
= mx + b, where m is the slope and b is the y-intercept.
Example 1: Find the slope-intercept form of an equation of the line that has a
slope of 1 and passes through (-1, 3). y = mx + b
2. Point-slope Form of a Linear Equation – The point-slope form of the equation
of a line is Y – y1 = m(X- x1) , where (x1 , y1) are the coordinates of a point on
the line and m is the slope of the line.
Example 2: Find the equation of the line that passes through (4, 9) and (2,
0).
Example 3: Write the equation of a line in slope intercept form containing
(1, 7) that is perpendicular to the line y =(1, 7) that is perpendicular to the line y
=- 1 x +1.
Example 4: An apartment complex charges $525 per month plus a $750 security
deposit. Write an equation to represent total annual cost A for r months of rent.
Skip This Section
Geometry 3-5 Proving Lines Parallel
A. Identify Parallel Lines
1. Postulate 3-4 -If two lines in a plane are cut by a transversal so that
corresponding angles are congruent, then the lines are parallel.
abbreviation -If corr <’s are , then lines are ||.
If <3 hen
2. Postulate 3-5 -If given a line and a point not on the same line, then there exists
exactly one line through the point that is parallel to the given line.
3. Theorem 3-5 -If two lines in a plane are cut by a transversal so that a pair of
alternate exterior angles is congruent, then the two lines are parallel.
abbreviation: If alt. ext. <s are , then the lines are ||
If <1  <8, then ______
4. Theorem 3-6 -If two lines in a plane are cut by a transversal so that a pair of
consecutive interior angles is supplementary, then the two lines are parallel.
abbreviation: If cons. int. <s are suppl., then the lines are ||.
If m<4 + m<6 = 180 o, then _____________
5. Theorem 3-7 -If two lines in a plane are cut by a transversal so that a pair of
alternate interior angles is congruent, then the two lines are parallel.
abbreviation: If alt. int. <s are , then the lines are ||
if <4 <5, then ___________
6. In a plane, if two lines are perpendicular to the same line, then they are
parallel.
abbreviation: If 2 lines are ┴ to the same line, then lines are ||.
If a ┴ t, and b ┴ t, then ______
Example 1: Determine which lines if any are parallel.
p
o
103
q
o
77
102o
r
Example 2: Find x and m<ZYN, so that PQ MN .
Geometry 3-6 – Perpendiculars and Distance
A. Distance from a point to a line
1. The distance from a line to a point not on the line is the length of the segment
perpendicular to the line from the point.
Example 1: Draw the segment that represents the distance from R to AB. (honors
by construction)
Example 2: Construct a line.
to line r through A(-4,0), which is not on r, then find the distance from A to .
B. Definition of the distance between parallel lines.
-The distance between two parallel lines is the distance between one of the lines
and any point on the other line.
Example 3: Find the distance between the parallel lines l and m whose equations
are y = 2x + 7 and y = 2x - 7, respectively.