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Review over Angle Relationships

So, before we really jump into how to begin a mathematical proof, it wouldn’t be a bad idea to review over what we have learned (because we are going to use it to prove things later on.)

So, let’s review over the angle relationships that we learned.

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SOME IMPORTANT DEFINITIONS TO GO REMEMBER

To make sure we don’t forget what we have learned, here are some definitions we need to review over just in case. This will also help us when we begin talking about the other different types of angles.

Parallel lines – Parallel lines are lines in a plane that are always the same distance apart, i.e., parallel lines are in the same plane and never intersect.

Transversal - In geometry, a transversal is a line that passes through two lines in the same place at two distinct points.

In other words, a transversal line passes through two lines, like so: (picture of parallel lines with a transversal)

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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: (insert the picture of the two parallel lines with a transversal and highlight the corresponding angles).

Okay, so why is this a thing?

Well, here’s why

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Vertical Angles

We now know about corresponding angles, but what about 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.

Here’s why

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Supplementary Angles

So, we actually already went over supplementary angles.

Supplementary angles are angles that when added together, equal 180 degrees.

So, when we have either a single line intersecting another line, or a transversal intersecting two parallel lines, the angles that are formed will be supplementary.

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ALTERNATE INTERIOR ANGLES

So, we’ve actually already seen alternate interior angles in the homework, and you actually already proved them, but let’s take a look at them anyway.

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.

So, I know that explanation doesn’t make that much sense, so something like this:

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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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NOW LET’S JUMP INTO PROOFS

Again, the review was important because we will be using these angle relationships in proving different theorems, which is sort of the foundation of actual mathematical calculation.

So without further ado…..

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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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SO, HOW DO WE WRITE THEM?

Well, think of a proof like a game of uno, and all of the postulates, definitions, and theorems that you know are different types of cards.

You use what you have to build on itself until you can prove what you want to prove.

So, for example:

Given:

Prove:

Using:

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So, we begin:

We start with what they gave us:

to eventually get:

So:

And use what we have

And now we can finally finish with:

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Proofs are incredibly similar

You are still given something to start with (as well as an endpoint), and you will use what you know to logically build the proof, stacking each idea on top of itself, so you can eventually prove whatever they are asking you to prove.

So, to show you what I mean, let’s prove that triangles always equal 180 degrees.

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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