Math Is Fun Forum

I had a lot of trouble with these last two questions, if it is possible I would like an answer with the explanation as I'm running out of time. The answers need to be separated into three points as 14a, 14b and 14c since there are 3 definitions/

    Imagine you have been called as a expert witness in a court case.  Your expertise is in the area of planes (not airplanes, just planes in geometry).  You have been asked the following questions.  Your task is to convince the jury that there is, in fact, a plane.  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.

14.  Question from the lawyer:  "Dr. Expert, I only see three parallel lines here.  Terry said that having these three parallel lines mean you have a plane.  I enter Exhibit A which shows three definitions of a plane.  From what I see, none of these definitions say that three parallel lines define a plane.  Explain how the definitions in Exhibit A prove that three parallel lines define a plane."   (This question is worth 3 points.)

15.  Question from the lawyer:  "Dr. Expert, I only see an angle between 0° and 180° here.  Kelly said that having this angle means you have a plane.  I enter Exhibit A which shows three definitions of a plane.  From what I see, none of these definitions say that an angle defines a plane.  Explain how the definitions in Exhibit A prove that an angle defines a plane."   (This question is worth 3 points.)

The definitions of an angle, plane and ray are:

A two-dimensional object of infinite proportions upon which you can have points, lines, angles, etc.

Think of it as a perfectly flat and unending sheet of paper.

A plane is defined by any of the following:

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

Remember that since a plane is perfectly flat, a line not on the plane that intersects with it would intersect at 1 point only (like an arrow through a sheet of paper would only go through 1 hole in the paper, even if the arrow goes through at an angle).

A ray is part of a line and is the set of points lying in a single direction from an endpoint.

In an intuitive way, a ray looks like ½ of a line, because it continues indefinitely in one direction, but has a definite endpoint in the other direction.  To define a ray, you need one endpoint, and then a second point somewhere along the ray (not the same as the endpoint) to define the direction. 

Rays are represented by the names of two points (the first must be the endpoint) with a single ended arrow over them.

We will represent rays using the letter "ry."  For example, ry_MB

An angle is a union of two rays having the same endpoint.

To define an angle, you need to be able to define both rays, and they need to have the same endpoint.

Angles are respresented by a “less than” sign (<) followed by the names of three points.  The endpoint of both rays is the middle point.  So the angle <BAC is the union of ry_AB and ry_AC.