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 &TF TTF FFF |TF TTT FTF p q ~q p -> qp & ~q TTFTF TFTFT FTFTF FFTTF 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 , #1-13. Test on 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 m  AVB + m  BVC = m  AVC A B C V

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.