Transcript Chapter 3

Chapter 3
Danny Ramsey, Ashley Krieg,
Kyle Jacobs, and Chris Runion
What is Ch. 3 About?
 It
is about lines and angles.
 We
learned:
 the
properties of parallel and perpendicular
lines.
 Six different ways to prove lines are
parallel.
 How to write an equation of a line with the
given characteristics
3.1
Lines and Angles
3.1 Vocabulary
Parallel lines: two lines that are coplanar
and never intersect
 Skew lines: two lines that are not coplanar
and never intersect
 Parallel planes: two planes that never
intersect

Review…
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RU || WZ
WZ and TY are skew
lines
Plane RUZW and
plane STYX are
parallel planes
Postulates

Parallel Postulate: If there is a line and a
point not on the line, then there is exactly
one line through the point parallel to the
given line
P
l
There is exactly one line through
point P parallel to l.
Postulates Continued

Perpendicular postulate: If there is a line
and a point not on the line, then there is
exactly one line through the point
perpendicular to the given line.
P
l
There is exactly one line through P
perpendicular to l.
Construction Activity
Perpendicular lines
Draw a line (l) and a Point (P) off of the line.
Put point of compass at P and open wide
enough to intersect l twice. Label those
intersections A and B. Using the same
radius, draw an arc from A and B. Label
the intersection Q. Use a straightedge to
draw PQ. PQ
l
More Vocabulary!
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Transversal: the line that intersects two or more
coplanar lines at different points
Corresponding Angles: angles that occupy
corresponding positions
Alternate Exterior Angles: two angles that are
outside the two lines on opposite sides of the
transversal
Alternate Interior Angles: two angles between
the two lines on opposite sides of the transversal
Consecutive Interior Angles: two angles that lie
between the two lines on the same side of the
transversal
VOCAB PICTURES
Transversal is RED
1
2
3
4
5
7
Corresponding Angles:
1&5
Alternate Exterior
Angles: 1 & 8
6
8
Alternate Interior Angles:
3&6
Consecutive Interior Angles:
3&5
3.2
Proof and Perpendicular Lines
3.2 Vocabulary

Flow Proof: uses arrows and boxes to
show the logical flow

Example
3.2 Theorems
If two lines intersect to form a linear pair of
congruent angles, then the lines are
perpendicular.
 If two sides of two adjacent acute angels
are perpendicular, then the angles are
complementary.
 If two lines are perpendicular, then they
intersect to form four right angles.

3.3
Parallel lines and Transversals
Corresponding Angles Postulate

If two parallel lines are cut by a
transversal, then the pairs of
corresponding angles are congruent
1
2
<1 = <2
3.3 Theorems

Alternate Interior
Angles: if two
parallel lines are
cut by a
transversal, then
the pairs of
alternate interior
angles are
congruent
3
4
<3 = <4
3.3 Theorems

Consecutive
Interior Angles: if
two parallel lines
are cut by a
transversal, then
the pairs of
consecutive interior
angels are
supplementary.
5
6
M<5 + M<6 = 180˚
3.3 Theorems

Alternate Exterior
Angles: if two
parallel lines are
cut by a
transversal, then
the pairs of
alternate exterior
angles are
congruent
7
8
<7 = <8
3.3 Theorems

Perpendicular
Transversal: if a
transversal is
perpendicular to
one of two parallel
lines, then it is
perpendicular to
the other
j
h
k
J
K
3.4
Proving Parallel Lines
POSTULATE

Corresponding angles converse: if two
lines are cut by a transversal so that
corresponding angles are congruent, then
the pairs of alternate interior angles are
congruent
j
k
J || k
Theorems about Transversals

Alternate Interior
Angles Converse: if
two lines are cut by a
transversal so that
alternate interior
angles are congruent,
then the lines are
parallel
j
3
1
If <1 = <3, then j || k
k
Theorems about Transversals

Consecutive Interior
Angles Converse: if
two lines are cut by a
transversal so that
consecutive interior
angles are
supplementary, then
the lines are parallel
j
2
1
If m<1 + m<2 = 180°, j || k
k
Theorems about Transversals

Alternate Exterior
Angles Converse: if
two lines are cut by a
transversal sot that
alternate exterior
angles are congruent,
then the lines are
parallel.
If <4 = <5, then j || k
j
4
k
5
3.5
Using Properties of Parallel Lines
Theorems

If two lines are
parallel to the same
line, then they are
parallel to each other.
r
p
q
If p || q and q || r, p || r

In a plane, if two lines
are perpendicular to
the same line, then
they are parallel to
each other
Construction Activity
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Copying an Angle
Draw an acute angle with the vertex A
Below the angle, draw a line using a straight edge put a
point on the line and label it D
Using a compass, put the point on A and open wide
enough to intersect both rays. Label the intersections B
and C
Using the same radius on the compass, draw an arc with
the center D, label the intersection E
Draw an arc with the radius BC and center E, label the
intersection F
Draw DF. <EDF = <BAC
Construction Activity
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Parallel Lines
Draw line M, using a straight edge, and point P off of the
line.
Draw points Q and R on line M. Draw PQ
Draw an arc with the center at Q so it crosses QP and
QR
Now copy <PQR, as shown in the previous construction
activity, on QP. Be sure the angles are corresponding,
Label the new angle <TPS
Draw PS. Since <TPS and <PQR are congruent
corresponding angles, PS || QR
3.6
Parallel Lines in the Coordinate
Plane
YOU MUST KNOW THIS!!!
RISE
SLOPE =
Y2 – Y1
RUN
=M
X2 – X1
y
Now, the picture
(X2 , Y1)
Y2 – Y1
RISE
(X1 , Y1)
X2 – X1
RUN
x
Slope of Parallel Lines Postulate

In a coordinate plane,
two nonvertical lines
are parallel is and
only if they have the
same slope. Any two
vertical lines are
parallel.
Slope = -1
SLOPES

Lines that have the same slope are
parallel. Y = 2x + 3 ; Y = 2x – 6
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Lines that are perpendicular have opposite
reciprocal slopes. Y = -2x + 3 ; Y = 1/2x -9
3.7
Perpendicular lines in the
coordinate plane
Slopes of Perpendicular Lines

In a coordinate plane,
two nonvertical lines
are perpendicular if
and only if the product
of their slopes is -1.
Vertical and horizontal
lines are
perpendicular.
Product of Slopes: 2 ( - ½ ) = -1
But Wait!!!
When Will I Ever Use This???
Sailing

There are three basic sailing maneuvers sailing into the wind, sailing across the wind,
and sailing with the wind. These three
maneuvers allow a sailboat to travel in
almost any direction. A boat that is sailing
into (or against) the wind is actually sailing
at an angle of about 45° to the direction of
the wind. A sailboat that is sailing into the
wind must follow a zigzag course called
tacking in order to avoid sailing directly into
the wind. When a boat is pointed directly
into the wind, the sails are rendered useless
and the boat loses its ability to move. A boat
can reach maximum speed by sailing across
the wind or reaching. In this situation, the
wind direction is perpendicular to the side of
the boat. The third sailing technique is called
sailing with the wind or running. Here, the
sail is almost at right angles with the boat
and the wind literally pushes the boat from
the stern
Graphic Artists

Graphic artists are creative,
analytical, and detail-oriented.
It is important to be able to
create a visual image of an
idea. This talent requires
strong spatial reasoning skills.
The use of various types of
graphic design software
involves an understanding of
geometric ideas such as
scaling and transformations,
and an understanding of the
use of percents in mixing
colors.
REAL WORLD PROBLEM
Apple St.
85 degrees
Are Apple and Orange
Streets Parallel?
Are there any
Perpendicular
Intersections?
Orange St.
90 degrees
Watermelon Ave.