Geometry Lesson: "Imagine you have been called as an expert witness in a court case...."

[jen.jen302]

Hi, I am having some trouble with my geometry lesson.

The question is:

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. 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 each definition proves that an angle defines a plane."

14. Definition 1:
15. Definition 2:
16. Definition 3:

The definitions 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

My Answers were:

14. Definition 1: An angle needs three co-linear points, which is one thing that defines a plane.
15. Definition 2: Any other point, not lying on the line would create an angle.
16. Definition 3: A plane is defined by two intersecting lines, which is the form of an angle. An angle also needs an intersecting point.

My teacher wants me to be much more specific about how the given angle proves there is a plane using each of the definitions. He also wants to know where the points are located and how an angle has lines. I am at a total loss and I have fallen back in this class, some help would be much appreciated.

 

[Dr.Peterson]

The definitions 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

My Answers were:

14. Definition 1: An angle needs three co-linear points, which is one thing that defines a plane.
15. Definition 2: Any other point, not lying on the line would create an angle.
16. Definition 3: A plane is defined by two intersecting lines, which is the form of an angle. An angle also needs an intersecting point.

My teacher wants me to be much more specific about how the given angle proves there is a plane using each of the definitions. He also wants to know where the points are located and how an angle has lines. I am at a total loss and I have fallen back in this class, some help would be much appreciated.

First, be very careful when you write in math. In 14, you presumably meant "non-collinear", right? Collinear points do not define a plane.

Furthermore, what does "needs" mean? What you need to do is to explain how, given the existence of the angle, you have all the elements of this definition of a plane. You probably need to start from your definition of an angle, which might be something like "the union of two rays with the same starting point". Since an angle can in fact be two rays that form a line, you may also need to use the fact that this is a 40 degree angle, not a 180 degree angle. You need to state how to find three non-collinear points in the angle, not just claim they are there; one might be the vertex, and the other two might be one on each ray. You may want to be very specific, and propose constructing a circle with radius 1 centered at the vertex, and use its intersections with the two rays as the other two points. That is what "specific" means in your teacher's instructions.

Essentially, you are learning how to write a mathematical proof, which is an argument that no one can find any holes in. Everything you say must be directly implied by something else you've said (including the "givens"). And the way to learn how to write a proof, just like the way you learned to write a persuasive essay, is by reading examples of good proofs, and then copying their style. I presume you have been given some; otherwise your teacher is asking you to do something without having seen a model of what it should look like. Look carefully at those models! And don't be afraid to ask your teacher to point out examples of proofs that don't have the problems that your statements do.

 

[pka]

Hi, I am having some trouble with my geometry lesson.
The question is: 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. 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 each definition proves that an angle defines a plane."
14. Definition 1:
15. Definition 2:
16. Definition 3:
The definitions 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
My Answers were:
14. Definition 1: An angle needs three co-linear points, which is one thing that defines a plane.
15. Definition 2: Any other point, not lying on the line would create an angle.
16. Definition 3: A plane is defined by two intersecting lines, which is the form of an angle. An angle also needs an intersecting point.
My teacher wants me to be much more specific about how the given angle proves there is a plane using each of the definitions. He also wants to know where the points are located and how an angle has lines. I am at a total loss and I have fallen back in this class, some help would be much appreciated.

These are questions very dear to me. Before retiring I taught axiomatic geometry (graduate/undergraduate) maybe forty times.
We build up to many concepts. Notation: \(\displaystyle A-B-C\) means that \(\displaystyle A.~B.~\&~C\) are three points and \(\displaystyle B\) is between \(\displaystyle A~\&~C\). If \(\displaystyle P~\&~Q\) are two points the \(\displaystyle \overline{PQ}=\{P,Q\}\cup\{X: P-X-Q\}\) is known as a line segment with end points \(\displaystyle P~\&~Q\).
Then \(\displaystyle \overrightarrow {AB}=\overline{AB}\cup\{X: A-B-X\}\) is a ray with endpoint \(\displaystyle A\).
Each of \(\displaystyle \overrightarrow {AB}~\&~\overrightarrow {AC}\) is a ray and the angle \(\displaystyle \angle BAC=\overrightarrow {AB}\cup\overrightarrow {AC}\)
Can you see that the measure of angle is \(\displaystyle 0\le m(\angle BAC) \le\pi~?\)
So it is possible to have a zero angle and a straight angle.

 

[Jj101]

Did you ever find the correct answers for questions 14-16? Please reply back!