Bellringer

• Block 2: Quizlets VENN and TRT. You have 5 minutes.• Blocks 1 & 3:

1. Write a logic table that you think describes p and q both being true at the same time (“AND”). Use the symbol ‘&’.

2. Write a logic table that you think describes at least one of them (p & q) being true (“OR”). Use the symbol ‘|’.

3. Write a truth table that finds the values for p -> q and p & ~q. Do you see a relationship? Are they equivalent?

Solutions are on the next slide.

Bellringer Solutions& T F

T T F

F F F

| T F

T T T

F T F

p q ~q p -> q p & ~q

T T F T F

T F T F T

F T F T F

F F T T F

p -> q and p & ~q aren’t equivalent – they’re 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.That’s the only time that p -> q is false.

Use Postulates and Diagrams

Section 2.4

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.

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

mAVB + mBVC = mAVCA

B

CV

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.

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 wouldn’t 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.

Postulate #6: Every line contains at least two points.

• Every line actually contains an INFINITE number of points.

• This postulate mentions only two because it’s a less strict requirement. – In Math, we try to keep the postulates as non-

restrictive as we can.

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.

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 we’re 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).

Postulate #9: A plane contains at least three noncollinear points.

• There are actually infinite points – we’re 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).

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.

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.

Solutions are on the next slide. No peeking!

Solutions are on the next slide.

Diagrams Lie!

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

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.

A 3-D Diagram

The solution is on the next slide.

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 don’t have such a line shown. We can’t assume.• Segment AB is shown with a perpendicular mark, so we know that it is plane S.• Segment CD doesn’t have such a mark; we can’t assume that it’s perpendicular to plane T.• The diagram clearly shows that lines AF and BC intersect at point B, so we know that is true.

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.