Bellringer

1
Bellringer

• Block 2 Quizlets VENN and TRT. You have 5
  minutes.
• Blocks 1 3
• Write a logic table that you think describes p
  and q both being true at the same time (AND).
  Use the symbol .
• Write a logic table that you think describes at
  least one of them (p q) being true (OR). Use
  the symbol .
• Write a truth table that finds the values for p
  -gt q and p q. Do you see a relationship? Are
  they equivalent?
• Solutions are on the next slide.

2
Bellringer Solutions
T F
T T F
F F F
T F
T T T
F T F
p q q p -gt q p q
T T F T F
T F T F T
F T F T F
F F T T F
p -gt q and p q arent equivalent theyre
opposites. Whenever one is true, the other is
false. This makes sense p q means that the
hypothesis is true, but the conclusion is
false. Thats the only time that p -gt q is false.
3
Use Postulates and Diagrams

• Section 2.4

4
Objectives Announcements

• Add new postulates to our repertoire
• Recognize the use of postulates in diagrams
• Diagram postulates
• HW for next time page 134-136, 1-13.
• Test on 2.1-2.4 next class! We will review
  before the test.

5
Old Postulates

• From Chapter 1
• Postulate 1 Ruler Postulate
• Postulate 2 Segment Addition Postulate
• A B C AB BC AC
• Postulate 3 Protractor Postulate
• Postulate 4 Angle Addition Postulate
• m?AVB m?BVC m?AVC

A
B
V
C
6
New Postulates (Point, Line, Plane)

• 5 Through every two points, there is exactly
  one line.
• 6 Every line contains at least two points.
• 7 If two lines intersect, their intersection is
  exactly one point.
• 8 Through any three noncollinear points, there
  exists exactly one plane.
• 9 A plane contains at least three noncollinear
  points.
• 10 If two points lie in a plane, then the line
  connecting them lies in the plane.
• 11 If two planes intersect, then their
  intersection is a line.

7
Postulate 5 Through every two points, there is
exactly one line.

• As with all postulates, this should be obvious.
• It lets us use the notation AB to refer to the
  line through points A and B. Without this
  postulate, we wouldnt know that there is such a
  line and there could be more than one.
• Example of more than one
• Look at the North and South poles of a globe.
• If we allow lines to be drawn on the sphere,
  there are many lines going from the North Pole to
  the South Pole (lines of longitude).
• Since we draw lines on planes instead of spheres,
  this does not happen.

8
Postulate 6 Every line contains at least two
points.

• Every line actually contains an INFINITE number
  of points.
• This postulate mentions only two because its a
  less strict requirement.
• In Math, we try to keep the postulates as
  non-restrictive as we can.

9
Postulate 7 If two lines intersect, their
intersection is exactly one point.

• P
• To see why we need this, remember the globe
• Any line going through the north pole would
  also go through the south pole.
• These are lines of longitude.
• All such lines would intersect in two points
  instead of one!
• We are drawing our lines on planes, so that
  cannot happen.

10
Postulate 8 Through any three noncollinear
points, there exists exactly one plane.

• Remember the triangle we created with string the
  first week of school? That triangle is part of
  the plane were talking about.
• If the three points were collinear, we could have
  many planes through them all in fact, an
  infinite number.
• This is a lot like Postulate 5 (through any two
  points there is exactly one line).

11
Postulate 9 A plane contains at least three
noncollinear points.

• There are actually infinite points were just
  trying to be non-restrictive again. (Three is a
  weaker requirement).
• The three points form a triangle in the plane.
• This is like Postulate 6 (a line contains at
  least two points).

12
Postulate 10 If two points lie in a plane, then
the line connecting them lies in the plane.

• We know that there is such a line because of
  Postulate 5.
• This is an example of why Postulate 5 is
  important.

13
Postulate 11 If two planes intersect, then
their intersection is a line.

• There was a question about this (along with a
  diagram) on the Chapter 1 test.
• Example
• The floor of the classroom intersects with the
  front wall of the classroom.
• Their intersection is the line along the bottom
  of that wall.

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Solutions are on the next slide. No peeking!
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Solutions are on the next slide.
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Diagrams Lie!

• Reminder Diagrams are often misleading. Here
  are some examples.

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Perpendicular Figures

• This is an example of where symbols such as the
  red right angle marker are important.
• Without them, we would not be able to assume that
  line t really is perpendicular to the plane.

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A 3-D Diagram
The solution is on the next slide.
21
A 3-D Diagram

• A, B, and F are collinear, since line AF is
  shown and B is on it in the diagram.
• E, B, and D are collinear dont have such a
  line shown. We cant assume.
• Segment AB is shown with a perpendicular mark,
  so we know that it is ? plane S.
• Segment CD doesnt have such a mark we cant
  assume that its perpendicular to plane T.
• The diagram clearly shows that lines AF and BC
  intersect at point B, so we know that is true.

22
Classwork

• You have a handout with the pages needed for this
  assignment.
• Do 3-8, 11-13, 14-23, 26, 29, 31, 32, 39, 42, 45.