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How to do a Proof

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Tatoranaki
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PostTatoranaki on Wed Mar 31, 2010 10:55 pm

[08:08:28 01/04/10] hotdogsaucer : What's a proof?

[08:08:30 01/04/10] @ Tatoranaki : Please prove that this triangle is a rectangle.

[08:08:33 01/04/10] @ Tatoranaki : Using theorems.

[08:08:36 01/04/10] @ Tatoranaki : And definitions.

[08:08:40 01/04/10] @ Tatoranaki : Well... let's see.

[08:08:46 01/04/10] @ Tatoranaki : The chatbox has four sides...

[08:08:50 01/04/10] @ Tatoranaki : They're equal right?

[08:08:52 01/04/10] @ Tatoranaki : WRONG!

[08:08:54 01/04/10] @ Tatoranaki : You don't know that.

[08:09:00 01/04/10] @ Tatoranaki : You have to prove it.

[08:09:03 01/04/10] @ Tatoranaki : But how?

[08:09:10 01/04/10] @ Tatoranaki : By Theorems! Cool

[08:09:28 01/04/10] hotdogsaucer : Ok...

[08:09:31 01/04/10] @ Tatoranaki : Lemme' find one.

[08:09:33 01/04/10] @ Tatoranaki : A proof.

[08:09:41 01/04/10] @ Tatoranaki : I'll you an example.

[08:09:49 01/04/10] hotdogsaucer : Ok

[08:09:54 01/04/10] hotdogsaucer : I'm watching

[08:09:54 01/04/10] @ Tatoranaki : Of how oh so fun they are... (voice is dripping with sarcasm)

[08:10:23 01/04/10] hotdogsaucer : ...

[08:10:31 01/04/10] @ Tatoranaki : Okay.

[08:10:35 01/04/10] @ Tatoranaki : Let's see.

[08:10:45 01/04/10] @ Tatoranaki : In every proof you have a "to prove" and a "given."

[08:10:50 01/04/10] @ Tatoranaki : The given tells you what you have.

[08:10:56 01/04/10] @ Tatoranaki : And is usually accompanied by a drawing.

[08:11:07 01/04/10] @ Tatoranaki : In a proof, it's like being in a courtroom.

[08:11:11 01/04/10] @ Tatoranaki : You must defend your case.

[08:11:25 01/04/10] @ Tatoranaki : THE CLIENT IS GUILTY! - Prosecutor

[08:11:31 01/04/10] @ Tatoranaki : Prove it. - Judge

[08:11:57 01/04/10] hotdogsaucer : ...

[08:12:10 01/04/10] @ Tatoranaki : Based on the DNA analysis taken at the scene of the crime, and the fingerprints and bullet marks, we have concluded it was Person #1. -Prosecutor

[08:12:16 01/04/10] @ Tatoranaki : Very well. -Judge

[08:12:32 01/04/10] @ Tatoranaki : So like a court case, you have to prove your argument.

[08:12:36 01/04/10] @ Tatoranaki : Which is the "to prove."

[08:12:42 01/04/10] hotdogsaucer : Got it

[08:13:01 01/04/10] @ Tatoranaki : Okay, one moment.

[08:13:06 01/04/10] @ Tatoranaki : Let's find a good one...

[08:13:30 01/04/10] @ Tatoranaki : One sec... lemme' dig in my horror-filled math book.

[08:13:40 01/04/10] hotdogsaucer : Take your time

[08:14:35 01/04/10] @ Tatoranaki : Here we go!

[08:14:41 01/04/10] @ Tatoranaki : Perfect, a 5-step proof.

[08:15:09 01/04/10] @ Tatoranaki : Given (which is your "evidence" that you are presented with at the beginning):

[08:15:43 01/04/10] @ Tatoranaki : Figure ABCD is a parallelogram, and M is the midpoint of segment AB.

[08:16:18 01/04/10] @ Tatoranaki : Prove (Your Case): If segment MD = MC then ABCD is a triangle.

[08:16:29 01/04/10] @ Tatoranaki : Okay, so since you can't see the picture, I'll describe it for ya'.

[08:16:38 01/04/10] @ Tatoranaki : You've got a "rectangle."

[08:16:47 01/04/10] @ Tatoranaki : With triangle in the middle.

[08:17:00 01/04/10] @ Tatoranaki : And naturally, that forms to other triangles.

[08:17:11 01/04/10] @ Tatoranaki : |/\\

[08:17:17 01/04/10] @ Tatoranaki : Like that sorta'.

[08:17:18 01/04/10] hotdogsaucer : I get it

[08:17:27 01/04/10] @ Tatoranaki : I mean... |/\|

[08:17:28 01/04/10] @ Tatoranaki : Okay...

[08:17:31 01/04/10] @ Tatoranaki : Have you done proofs?

[08:17:36 01/04/10] hotdogsaucer : Nope

[08:17:42 01/04/10] hotdogsaucer : Is this high school stuff

[08:17:48 01/04/10] @ Tatoranaki : Ah, alright, moving on. (Yup...)

[08:17:59 01/04/10] @ Tatoranaki : Now, first we need to prove this.

[08:18:01 01/04/10] hotdogsaucer : Oh..... Sad

[08:18:14 01/04/10] @ Tatoranaki : By presenting statements, and giving evidence for them.

[08:18:27 01/04/10] @ Tatoranaki : So every time you make a statement, you need to give a "justification."

[08:18:38 01/04/10] @ Tatoranaki : Step 1 is always presenting the given.

[08:18:48 01/04/10] @ Tatoranaki : Afterall, it's evidence that was given to you, so why not?

[08:18:51 01/04/10] @ Tatoranaki : It helps your case.

[08:19:01 01/04/10] @ Tatoranaki : So step zero looks like this.

[08:19:08 01/04/10] @ Tatoranaki : *I mean step 0, not step 1

[08:19:39 01/04/10] @ Tatoranaki : 0. ABCD is a Parallelogram, and M is the midpoint of segment AB.

[08:19:43 01/04/10] @ Tatoranaki : Then we have to prove it.

[08:19:49 01/04/10] @ Tatoranaki : 0. Given

[08:20:06 01/04/10] hotdogsaucer : Ok

[08:20:10 01/04/10] @ Tatoranaki : Next, we go onto step 1, which is where we start proving that ABCD is a rectangle.

[08:20:21 01/04/10] @ Tatoranaki : We need to prove each part, one by one.

[08:20:33 01/04/10] @ Tatoranaki : 1. Segment AM=MB

[08:20:41 01/04/10] @ Tatoranaki : Now you have various proofs and definitions.

[08:20:46 01/04/10] hotdogsaucer : ?

[08:21:06 01/04/10] @ Tatoranaki : And you must utilize your (evidence) proofs, to prove your statement.

[08:21:26] @ Tatoranaki : Here's a visual

[08:21:33] @ Tatoranaki : A_M_B

[08:21:44] @ Tatoranaki : |/\|

[08:21:52] @ Tatoranaki : D C

[08:21:53] hotdogsaucer : I get it!

[08:22:04] @ Tatoranaki : Yup. (combine the pictures together)

[08:22:07] hotdogsaucer : Wait, what's the DC?

[08:22:07] @ Tatoranaki : Moving on...

[08:22:17] @ Tatoranaki : The bottom.

[08:22:24] hotdogsaucer : Thanks

[08:22:26] @ Tatoranaki : Of the rectangle (and the center triangle)

[08:22:28] @ Tatoranaki : uh-huh.

[08:22:40] @ Tatoranaki : So we prove step 1 with... 1. Definition of a Midpt.

[08:22:50] hotdogsaucer : Midpt?

[08:23:20] @ Tatoranaki : Since that particular definition proves that "M" in our figure (midpoint, abbreviated) is infact a mid point.

[08:23:47] @ Tatoranaki : Step 2, we need to prove both opposite sides are equal.

[08:23:55] @ Tatoranaki : 2. AD = BC

[08:24:01] @ Tatoranaki : (prove it now...)

[08:24:25] hotdogsaucer : Keep going

[08:24:26] @ Tatoranaki : 2. Parallelograms have Opposite Sides Congruent

[08:24:42] @ Tatoranaki : (since it can also be seen as a parallelogram, by definition)

[08:25:11] hotdogsaucer : So rectangles can be called parallelograms?

[08:25:26] @ Tatoranaki : Yup.

[08:25:33] @ Tatoranaki : Since they have the same properties.

[08:25:39] @ Tatoranaki : All sides are parallel, right?

[08:25:44] @ Tatoranaki : So it's a parallelogram.

[08:25:50] hotdogsaucer : Yeah

[08:26:36] @ Tatoranaki : Just like a square is a rectangle, kite, parallelogram, trapezoid, isosceles trapezoid, kite, rhombus, and quadrilateral.

[08:26:47] @ Tatoranaki : Since it fits all those requirements.

[08:26:55] hotdogsaucer : ???

[08:27:13] hotdogsaucer : Don't squares have 4 congruent sides?

[08:27:17] @ Tatoranaki : Mhm.

[08:27:19] @ Tatoranaki : That's right.

[08:27:22] @ Tatoranaki : So it's all that.

[08:27:25] hotdogsaucer : And rectangles don't?

[08:27:30] @ Tatoranaki : Not all those are squares.

[08:27:36] @ Tatoranaki : A square is all those.

[08:27:41] @ Tatoranaki : But all those are NOT squares.

[08:27:44] @ Tatoranaki : It's a one-way thing.

[08:27:54] hotdogsaucer : Oh, I got that mixed up

[08:27:59] @ Tatoranaki : A square has everything a rectangle, trapezoid, parallelogram, etc. have.

[08:28:00] @ Tatoranaki : Mhm.

[08:28:02] hotdogsaucer : Thanks

[08:28:06] @ Tatoranaki : So next we want to prove the two triangles (opposite sides of the rectangle |/ \| ) are congruent.

[08:28:20] @ Tatoranaki : So, let's state that.

[08:28:42] @ Tatoranaki : 3. Triangle AMD is congruent to triangle BMC.

[08:28:48] @ Tatoranaki : Now we have to prove it...

[08:29:02] @ Tatoranaki : Which we can with SSS... which is the Side Side Side Congruence Theorem.

[08:29:42] @ Tatoranaki : And...

[08:29:51] hotdogsaucer : What's the SSS?

[08:29:54] @ Tatoranaki : The reason that is so...

[08:30:01] @ Tatoranaki : (I'm getting to that)

[08:30:14] hotdogsaucer : Sorry

[08:30:54] @ Tatoranaki : The SSS (Side Side Side Congruence Theorem) states: If, in two triangles, three sides of one are congruent to three sides of the other, then the triangles are congruent.

[08:31:29] hotdogsaucer : Thanks

[08:31:45] @ Tatoranaki : So in other words, we've been taking apart this "rectangle" (we're proving it's a rectangle), and making each part of it equal.

[08:32:25] @ Tatoranaki : Because if one part of it fails to meet the requirements, it's not a rectangle, even if it appears to be.

[08:32:35] @ Tatoranaki : Gotta prove those angles are equal.

[08:32:56] @ Tatoranaki : I mean...

[08:33:05] @ Tatoranaki : 1 sec, might as well tell you what we're trying to do...

[08:33:10] @ Tatoranaki : Prove it's a rectangle right?

[08:33:14] @ Tatoranaki : What's a rectangle?

[08:33:18] hotdogsaucer : Right

[08:33:41] hotdogsaucer : A rectangle has

[08:33:43] hotdogsaucer : 4 sides

[08:33:53] hotdogsaucer : 2 pairs of congruent sides

[08:34:24] hotdogsaucer : Right?

[08:34:51] hotdogsaucer : Or is there some other complicated way to put it?

[08:35:01] @ Tatoranaki : Always a complicated way. Razz

[08:35:15] @ Tatoranaki : Lemme' just din it.

[08:35:24] @ Tatoranaki : *find

[08:35:36] hotdogsaucer : Sad It's been quite complicated during this chat

[08:35:43] hotdogsaucer : ....

[08:35:49] hotdogsaucer : The chatbox is a rectangle, right?

[08:35:57] @ Tatoranaki : "A quadrilateral is a rectangle if and only if it has four right angles."

[08:35:58] hotdogsaucer : It looks like one...

[08:36:01] @ Tatoranaki : That's all it needs bud,.

[08:36:04] @ Tatoranaki : Just four right angles.

[08:36:10] hotdogsaucer : Ok

[08:36:17] @ Tatoranaki : If you prove it, then it is. Razz

[08:36:18] hotdogsaucer : Just four right angles

[08:36:29] hotdogsaucer : Now that's some useful info

[08:36:38] @ Tatoranaki : Uh-huh, which make other things, like congruence.

[08:37:00] hotdogsaucer : Congruese is being equal in size and shape, right?

[08:37:00] @ Tatoranaki : Now this proof we're doing... has 9 steps.

[08:37:31] @ Tatoranaki : So how about I just go and finish it off, and if you get too confused, you just ask?

[08:37:48] hotdogsaucer : Sure, that'd be great

[08:38:49] hotdogsaucer : Tat?

[08:39:33] @ Tatoranaki : Typing it up.

[08:39:43] hotdogsaucer : Ok, I can wait

[08:39:50] @ Tatoranaki : 4. The Measurement of Angle A (the top left-hand edge of the rectangle) is congruent to the measurement of angle BCD (the right-hand, backwards L shape of the rectangle),

[08:40:01] @ Tatoranaki : and the measurement of angle B is congruent to that of ADC.

[08:40:05] hotdogsaucer : Yes

[08:40:22] @ Tatoranaki : And the justification:

[08:40:26] hotdogsaucer : and the measurement of angle C is congruent to that of ABD?

[08:40:43] @ Tatoranaki : We're not stating that yet. Wink

[08:40:51] @ Tatoranaki : 9 step proof. o_O

[08:40:54] hotdogsaucer : ?

[08:41:03] @ Tatoranaki : We need to state one thing at a time.

[08:41:07] @ Tatoranaki : And only if they are relative.

[08:41:12] @ Tatoranaki : *one idea

[08:41:19] hotdogsaucer : Relative?

[08:41:22] @ Tatoranaki : So to prove that...

[08:41:30] @ Tatoranaki : (make sense in other words)

[08:42:17] @ Tatoranaki : So saying triangle ABC is congruent to CBM, and segment 1 is congruent to segment 2, and the angle 3 is a right angle, makes no sense.

[08:42:33] @ Tatoranaki : Okay, so to prove step for:

[08:42:35] @ Tatoranaki : *four

[08:42:47] hotdogsaucer : Wait, then why did you state that B's angle was congruent to ACD

[08:42:56] @ Tatoranaki : 4. Parallelogram Opposite Sides are Congruent

[08:42:59] hotdogsaucer : and that A's angle was congruent to BCD?

[08:43:05] hotdogsaucer : In one step?

[08:43:06] @ Tatoranaki : (It proves the different parts of the rectangle.)

[08:43:33] @ Tatoranaki : (Then you can do transitive property, which takes two different "ideas" and "fuses" them together, in the next step.)

[08:43:43] @ Tatoranaki : Okay, here's an example:

[08:43:44] hotdogsaucer : Then why didn't you mention angle C and D's congruence?

[08:43:54] @ Tatoranaki : I'll give you an example so you'll get it. Wink

[08:44:01] hotdogsaucer : THanks

[08:44:34] @ Tatoranaki : (this doesn't make sense in human terms, but it does in math terms)

[08:44:45] @ Tatoranaki : (just so you know, before you disagree.)

[08:44:52] hotdogsaucer : Ok

[08:44:54] @ Tatoranaki : Sally likes George.

[08:45:02] @ Tatoranaki : And George likes Mike.

[08:45:05] hotdogsaucer : Razz

[08:45:18] @ Tatoranaki : So by the transitive property, Sally likes Mike.

[08:45:18] hotdogsaucer : NOw that's funny!

[08:45:49] @ Tatoranaki : Got it?

[08:45:51] @ Tatoranaki : XD

[08:46:08] hotdogsaucer : Wait, how does Sally like Mike if George likes Mike and Sally liking Mike wasn't previously mentioned?

[08:46:21] hotdogsaucer : And Sally likes George instead?

[08:46:34] @ Tatoranaki : Because it makes sense in math terms! lol

[08:46:52] @ Tatoranaki : This isn't a psychological exercise, and examining of human relationships.

[08:47:05] hotdogsaucer : So, they're all connected in some way?

[08:47:13] hotdogsaucer : Mathematcially

[08:47:23] @ Tatoranaki : We're saying that if Sally likes George, and George likes Mike, you can conclude that Sally likes Mike, because she likes George who likes Mike.

[08:47:31] @ Tatoranaki : Get it? Got it? Good. Razz

[08:47:55] @ Tatoranaki : So in math terms:

[08:48:07] hotdogsaucer : Sort of like meeting a friend of a friend?

[08:48:44] @ Tatoranaki : If angle (A) is congruent to angle (B), and angle (B) is congruent to angle (C), by the transitive property, angle (A) is congruent to angle (C).

[08:48:57] @ Tatoranaki : Get it now?

[08:49:00] hotdogsaucer : Got it!

[08:49:03] hotdogsaucer : Razz

[08:49:09] @ Tatoranaki : Haha, there we go!

[08:49:17] @ Tatoranaki : Gotcha' doing Geometry already.

[08:49:35] hotdogsaucer : What grade math is that?

[08:49:38] @ Tatoranaki : How about I post the rest of the proof with a picture on the forum later? (or tmorrow)

[08:49:45] @ Tatoranaki : 11th Grade, Sophomore.
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Tatoranaki
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PostTatoranaki on Wed Mar 31, 2010 11:02 pm

[09:00:27 01/04/10] hotdogsaucer : 40 minutes.......... out of my Spring Break

[09:00:33 01/04/10] hotdogsaucer : BUT

[09:00:44 01/04/10] hotdogsaucer : That was 40 minutes used wisely!

[09:00:46 01/04/10] @ Tatoranaki : (drawing...)

[09:00:49 01/04/10] @ Tatoranaki : XD

[09:00:52 01/04/10] @ Tatoranaki : Yup.

[09:00:53 01/04/10] hotdogsaucer : For I learned about proofs!
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PostTatoranaki on Wed Mar 31, 2010 11:36 pm

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PostBarsky on Fri Apr 02, 2010 9:41 pm

Awesome. And I thought you didn't like math.
Anyway, I remember doing proofs in geometry two years ago; they were really fun.
Besides SSS, the other Congruence Theorems are SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side) and RHS (Right-angle-Hypotenuse-Side). Very fun stuff.
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