Math Is Fun Forum

Imagine you have been called as an expert witness in a court case.  Your expertise is in the area of planes (not airplanes, just planes in geometry).  Your task is to convince the jury that there is, in fact, a plane based on the given information.  You must prove all three of the definitions of a plane given in Lesson 1. You may need to include some other definitions such as the definition of an angle, a ray, etc.

Question from the lawyer:  "Dr. Expert, I only see a 70° angle here, Exhibit A. Kelly said that having this angle means you have a plane.  From what I see, none of the definitions of a plane say that an angle defines a plane.  Explain how each definition proves that an angle defines a plane."
Exhibit A: A plane can be defined by two lines that intersect at a point, forming an angle (the 70° one).

State the definition and then explain how you can prove each definition given the angle.

14. Definition 1:
      Proof:

15. Definition 2:
      Proof:

16. Definition 3:
     Proof:

The definitions given are:
- three points that are not collinear
- a line and a point not lying on the line
- two lines which intersect in a single point or are parallel

I just cannot wrap my head around this at all.

Here are my answers:
Definition 1: An angle requires 3 non-collinear points, which is one of the things that defines a plane.
Definition 2: A point not on the line would make an angle.
Definition 3: A plane can have 2 intersecting lines, which forms an angle.

The teacher said: "Now, you have said that an angle is made up of two rays having the same endpoint.
Definition 1: How does having two rays with the same endpoint tell us we have 3 noncollinear points?
Definition 2: How does having two rays with the same endpoint tell us we have a line and a point not on the line?
Definition 3: How does having two rays with the same endpoint tell us we have two lines that intersect?"

What am I supposed to write? I can't proceed to the next set of lessons if I don't finish this one, and I'm on a schedule. Please help.

A little context:
Statement: p ⟹ q
Converse: q ⟹ p
Inverse: not p ⟹ not q
Contrapositive: not q ⟹ not p

The complex statement is "If x^2 > 10, then x > 0."
7. "x = - 4" would be an example of a
a. converse
b. counterexample
c. contrapositive
d. counterintuition
e. counterpositive
f. counter

The complex statement is "Cars can take you everywhere."
10. "A car can't take you to the moon" would be the
a. converse
b. countermove
c. contrapositive
d. counterexample
e. counterpositive
f. counter

The complex statement is "Baseball players are athletes."
13. Which of the following is accurate?
a. the inverse of the statement is "If someone is a baseball player then someone is an athlete."
b. the statement is "If someone is an athlete, then they are a baseball player."
c. the statement can never be true.
d. baseball players all have great teeth and gums.
e. the inverse of the statement is not true.
f. the converse is: "Joey is a baseball player, and he is not an athlete."

Thank you!