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SO WHAT IS A PROOF?

Other than something that each student has been dreading all semester, a proof is:

“A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.”

Or, in other words, a mathematical proof is a logical mathematical argument used to show the truth of a mathematical statement.

So, basically, we know something works, but that’s not enough, we need to prove that it works.

This is done because in the math, everyone is skeptical of everything (much like the science world).

So, to make sure something works in mathematics, we have to show that it’s legit.

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Proof Example

Statements	Reason

Definition of Midpoint

SSS Congruence

Given

This time, let’s try to do it without the Uno cards.

So, what do we know?

Well, we know what we were given.

Definition of Midpoint

So what else do we know?

Well, we know that C is the midpoint of AE

But what does that mean?

Well, we know that since C is the midpoint of AE, C splits AE into 2 equal segments, AC, and CE (since this is the definition of a midpoint.)

We also know that since C is also the midpoint of BD, C splits BD into 2 equal segments, BC, and DC (again, the definition of a midpoint.)

And now, we have three sides that are congruent to three sides.

What postulate can we use there?

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THE TRIANGLE SUM CONJECTURE PROOF

Prove: The sum of the angles of

FEG = 180 degrees

Statements	Reason

Definition of angles residing on a line

Alternate interior Angles

Substitution

Thus:

The sum of the angles of

FEG = 180 degrees

Given

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Strategies to proofs

So, when it comes to proofs, there are a few things that we need to go over.

First, it’s important to know what you have that you can work with.

-This is where a journal of all of the theorems, postulates, and definitions can come in handy.

Second, come up with a plan

-Start with what they gave you

-Think about how you will get to the end of your proof. What are you trying to prove? What steps can you take to get there?

Suggestions:

Proofs can be pretty tough, but one thing that can help is instead of trying to prove whatever they are asking you to prove from the start (or what they gave you) to the end (or what they want you to prove), sometimes it helps to start at the end, and work backwards.

Write it out – explain, in your own words, how you know it works. Sometimes seeing it in your own writing can help you convert it to a formal proof.

Don’t stress – when you allow a proof to stress you out, it immediately becomes 10 times harder. If you need, take a break. Let your brain absorb what you need to prove it.

Talk to your teacher – emailing me for help in the next step can be beneficial as well. Mathematicians do this all of the time. They don’t email a teacher, per say, but they do get their peers to chime in and act as a second pair of eyes.

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SSS Example 1

Statements	Reason

Reflexive Property

SSS Congruence

Given

So, this may look weird, but try to look at this like the uno proofs (if that helps)

So, imagine we start with:

And we want:

And what we have to work with is:

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Example 2

Statements	Reason

Reflexive Property

SSS Congruence

Given

One more time, let’s look at this like an Uno proof

So, we are given:

And we want:

And what we have to work with is:

Definition of Midpoint

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MOVING ON

So, of course, since mathematics is a very big subject, we need to look into other proofs that can be proved.

Now, we’ve gone over Side Side Side congruence proofs, but now let’s move into something a little more advanced:

Side Angle Side Congruence proofs.

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PROVING SIDE ANGLE SIDE

So, Side Angle Side Proofs are similar to Side Side Side proofs, except we now have the addition of an angle, which means we have many more tools at our disposal.

For Side Side Side, we only had the ability to prove whether or not segments are congruent (which basically means reflexive property or midpoint definition.)

However, since we now have an angle included, we have all of the angle relationships that we have been studying.

So, let’s review over them really quick.

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Corresponding Angles

Alright, let’s start off with the definition of corresponding angles.

Corresponding angles – when two lines are crossed by a transversal, the angles in matching corners are called corresponding angles.

So, something like this:

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Vertical Angles

Vertical angles, by definition, are a pair of non-adjacent angles formed when two lines intersect.

So, in plain English, basically when two lines cross, they create these two angles.

These two angles are always congruent.

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ALTERNATE INTERIOR ANGLES

Alternate Interior Angles - If two parallel lines are transected by a third line (a transversal), the angles which are inside the parallel lines and on alternate sides of the third line are called alternate interior angles.

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Or, again, in plain English, if you have two parallel lines intersecting with a transversal, the angles on the inside are equal.

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ALTERNATE EXTERIOR ANGLES

Similar to Alternate Interior Angles, we’ve already seen Alternate Exterior Angles.

So, as is tradition, let’s start with the definition.

Alternate Exterior Angles - Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.

Again, in English, this means that when you have two parallel lines intersecting with a transversal, the angles on the outside are equal.

Again, we’re looking at something like this:

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So, now that we’ve reviewed, we can get into it

Again, Side Angle Side is more complicated than Side Side Side.

Now that we’ve reviewed over what we know, we can use that to help prove some of these problems.

So, let’s jump in!

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Example 1

Given: BC is congruent to CE

C is the midpoint of AD

BE intersects AD

Statements	Reason

BC is congruent to CE

C is the midpoint of AD

BE intersects AD

Vertical Angles

SAS Congruence

Given

First, let’s try this out with the uno cards (only one).

And we want:

And what we have to work with is:

Definition of Midpoint

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Example 2

Given: AC is a transversal between two parallel lines DB and EC

B is the midpoint between CA

DB is congruent to EC

Statements	Reason

AC is a transversal between two parallel lines DB and EC

B is the midpoint between CA

DB is congruent to EC

Corresponding Angles

SAS Congruence

Given

Alright, now let’s try this out without the Uno cards

So, first, we need to put the Given down.

So:

Definition of Midpoint

Okay, now, what do we know?

We know that AC is a transversal intersecting DB and EC

So, what do we know about angles when dealing with transversals that may help us out some?

Corresponding angles!

So we know, from corresponding angles, that

We also know that since B is the midpoint between CA, it creates two congruent segments: BA and CB

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that:

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Example 3

Given: DA is congruent to CB

DA is parallel to CB

AC transverses DA and CB

Statements	Reason

DA is congruent to CB

DA is parallel to CB

AC transverses DA and CB

Alternate Interior Angles

SAS Congruence

Given

Alright, so again, we need to start by putting what they gave us down.

So:

Reflexive Property

Okay, now, what do we know?

We know that AC is a transversal intersecting DA and CB

So, what do we know about angles when dealing with transversals that may help us out some?

Alternate Interior angles!

So we know, from alternate interior angles, that

We also know that we need a final congruent side to make this work.

However, due to the reflexive property, we know that AC is congruent to AC.

So:

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that:

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Example 4

Given: CE is parallel to FB

AD transverses CE and FB

FB is congruent to CE

DE is congruent to AF

Statements	Reason

CE is parallel to FB

AD transverses CE and FB

FB is congruent to CE

DE is congruent to AF

Alternate Exterior Angles

SAS Congruence

Given

Alright, so again, we need to start by putting what they gave us down.

So:

Now, the given gave us a ton of info

Most of the work is done here

We have two sides that are congruent, but to use SAS, we need to prove the angle between those two sides.

However , we do know that CE || FB, and AD transverses them

So what angle relationship can help us out here?

Alternate Exterior Angles!

So, because of alternate exterior angles, we know that

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that: