Danny Ramsey, Ashley Krieg, Kyle Jacobs, and Chris Runion

1
Chapter 3

• Danny Ramsey, Ashley Krieg, Kyle Jacobs, and
  Chris Runion

2
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
3.1

• Lines and Angles

4
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

5
Review

• RU WZ
• WZ and TY are skew lines
• Plane RUZW and plane STYX are parallel planes

6
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.
7
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.
8
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

9
More Vocabulary!

• 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

10
VOCAB PICTURES
Transversal is RED
1
2
3
4
5
6
8
7
Corresponding Angles 1 5 Alternate Exterior
Angles 1 8
Alternate Interior Angles 3 6 Consecutive
Interior Angles 3 5
11
3.2

• Proof and Perpendicular Lines

12
3.2 Vocabulary

• Flow Proof uses arrows and boxes to show the
  logical flow
• Example

13
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.

14
3.3

• Parallel lines and Transversals

15
Corresponding Angles Postulate

• If two parallel lines are cut by a transversal,
  then the pairs of corresponding angles are
  congruent

1
2

16
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

17
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
18
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

19
3.3 Theorems
j

• Perpendicular Transversal if a transversal is
  perpendicular to one of two parallel lines, then
  it is perpendicular to the other

h
k
J K
20
3.4

• Proving Parallel Lines

21
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
22
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
k
If
23
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
k
2
1
If m
24
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.

j
4
k
5
If
25
3.5

• Using Properties of Parallel Lines

26
Theorems

• 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

r
q
p
If p q and q r, p r
27
Construction Activity

• 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.

28
Construction Activity

• 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 construction activity, on QP. Be sure the angles
  are corresponding,
• Label the new angle
• Draw PS. Since corresponding angles, PS QR

29
3.6

• Parallel Lines in the Coordinate Plane

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YOU MUST KNOW THIS!!!
RISE RUN
SLOPE
Y2 Y1 X2 X1
M
31
Now, the picture
y
(X2 , Y1)
Y2 Y1 RISE
(X1 , Y1)
X2 X1 RUN
x
32
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
33
SLOPES

• Lines that have the same slope are parallel. Y
  2x 3 Y 2x 6
• Lines that are perpendicular have opposite
  reciprocal slopes. Y -2x 3 Y 1/2x -9

34
3.7

• Perpendicular lines in the coordinate plane

35
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
36
But Wait!!!

• When Will I Ever Use This???

37
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

38
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.

39
REAL WORLD PROBLEM
85 degrees
Apple St.
Are Apple and Orange Streets Parallel? Are there
any Perpendicular Intersections?
Orange St.
90 degrees
Watermelon Ave.