## How to do a Proof

on Wed Mar 31, 2010 10:55 pm

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**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!**[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.....[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.**[08:35:15]

**@****Tatoranaki****:****Lemme' just din it.**[08:35:24]

**@****Tatoranaki****:*****find**[08:35:36]

**hotdogsaucer****:****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.**[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.**[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.**[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****:**[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.**[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****:**[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.**## Testimonials

on Wed Mar 31, 2010 11:02 pm

[09:00:27 01/04/10]

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

[09:00:44 01/04/10]

[09:00:46 01/04/10]

[09:00:49 01/04/10]

[09:00:52 01/04/10]

[09:00:53 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!**## Re: How to do a Proof

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.

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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