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simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

I'm really at my wit's end with these geometry questions

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: I'm really at my wit's end with these geometry questions

hi simonmagusflies

It sounds like you have just missed some steps.  In this module, you have to be able to work systematically from the definitions.

We're told we have an angle.  So what is the definition for that?

The teacher said: "Now, you have said that an angle is made up of two rays having the same endpoint.

Is that the given definition of an angle?  I don't know because I'm not doing this course and CompuHigh seem to make up their own definitions.

If it is then you can prove you have three points, because the 'same endpoint' is one; another point on each ray gives you two more.

I'll be able to help better if you post the definitions exactly as given in the course notes.

Please answer in this format:

<geometric type> is defined by <these properties>

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: I'm really at my wit's end with these geometry questions

Hello, Bob, thanks for answering.

The definition of an angle is given as: "the union of two rays having the same endpoint."
A ray is defined as: "part of a line and is the set of points lying in a single direction from an endpoint."
A line segment is defined as: "the set of all points that lie between two selected points on a line."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: I'm really at my wit's end with these geometry questions

hi

The definition of an angle is given as: "the union of two rays having the same endpoint."
A ray is defined as: "part of a line and is the set of points lying in a single direction from an endpoint."
A line segment is defined as: "the set of all points that lie between two selected points on a line."

We have an angle.

=> we have two rays having the same endpoint.  I'll call that point A.

=> For the first ray there are a set of points with A as the endpoint.  Call any of the other points B.

Similarly for the second ray, I'll call one of its other points C.

Thus we have A, B and C; three distinct points, that are not collinear, so we have a plane.

You can also use either of the other definitions of a plane to reach the same conclusion.

Hope that keeps your teacher happy.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: I'm really at my wit's end with these geometry questions

Thank you very much, you've helped me out a lot!!

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: I'm really at my wit's end with these geometry questions

Sorry to bother, but if you have time, the teacher sent my work back with this comment: "You're on the right track with this thought! The definition of an angle does not include any mention of lines, so you will need to explain where any lines you need come from using what you learned in Lesson 1. Make sure to write a separate proof for each problem as well."

????? I'm dying here!! Can't move on to other lessons until I finish this one

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: I'm really at my wit's end with these geometry questions

I don't understand what your teacher is objecting to. Please post what you submitted.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

simonmagusflies

Member

Registered: 2021-05-23

Posts: 32

Re: I'm really at my wit's end with these geometry questions

I decided to give it one go and wrote "I have two rays with the same endpoint, which I'll call A. For the first ray, there's a set of points that uses A as an endpoint. Any other point I'll call B.

For the second ray it's about the same, but I'll call any other points C.

So we have A, B, and C, which are three non-collinear distinct points, so we have a plane. That's a point not on the line as well (I guess), and the lines intersect at A."

The teacher replied: "You're on the right track with this thought! The definition of an angle does not include any mention of lines, so you will need to explain where any lines you need come from using what you learned in Lesson 1. Make sure to write a separate proof for each problem as well."

Ahh I wish she'd just mark me off so I can move on!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: I'm really at my wit's end with these geometry questions

Oh I'm so sorry. My fault. Over two thousand years ago a Greek mathematician called Euclid developed the geometry that has been used ever since. He defined the line but had no use for rays. So I'm not used to this new concept.

Angles are defined in your course using rays but planes using lines. So we need to bridge the gap between the two.
After saying you have two rays we need to get to being able to say we have two lines. I'm hoping you can say that as a ray exists then a line also exists, as a ray is defined as part of a line.

Do this twice and you have two lines. As they are distinct except for the common point A, any other point on each line can be used to get three nonlinear points.

I'm hoping that's enough to complete the proof.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob