Proof Concepts: Definitions | It All Adds Up

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Today, we’ll discuss a concept found in proofs of all kinds. Definitions can be used to identify almost any aspect of geometry, and constitute an important step in solving a proof.

But first, some necessary background information:

Since this year, my focus is on discussing proofs and specifically the factors necessary to complete proofs, I’ve jumped forward through a few small, unnecessary units that involve other geometric topics. However, if you’d like to complete some research of your own, these are the topics that were covered:

  • Identifying and labeling polygons (ie dodecagons)
  • Writing conjectures involving a hypothesis, conclusion, and negations
  • Writing other forms of a conditional statement (ie the Contrapositive)
  • Law of Detachment & Syllogism
  • Basic geometric postulates (not used in proofs directly)
    • ie “through any two points passes exactly one line”

A definition is as simple as it sounds. Any geometric aspect has a definition. For example, a definition of a right angle is that it has a measure of 90 degrees. The definition of a pair of Vertical Angles involves two angles that intersect and share a vertex.

These definitions are incredibly useful, though, for reasons other than understanding. If a point serves as a midpoint of a line, then we can assume that the two segments created are congruent. But how would we represent this in terms of a proof? Well, as a reason, we would use the Definition of a Midpoint, because we know that a midpoint is a point that divides a line into two congruent segments.

In addition, definitions serve as excellent transition points to using theorems and postulates. The Vertical Angles Congruence Theorem states that a pair of vertical angles are congruent. In order for this to work, though, the angles have to be vertical. And how would you know that the angles are vertical? By the Definition of Vertical Angles. 

In terms of basic geometry, a definition will almost always precede a postulate or theorem, unless not such definition exists. For example, using the Angle Addition Postulate, there is nothing that we need to prove in order to use the postulate. There are no requirements, such as “they must share a vertices”. They just have to be any two angles. In this case, you can just say that the sum of two smaller angles equals that of the larger angle with the A.A.P. as the reason, with no definition. Of course, though, this isn’t always the case.

I get that, based on the way I am explaining this, it appears to be complicated. But when you start implementing definitions into simple proofs, they are a lot easier to understand. For example, take a look at this proof:

screen-shot-2018-02-20-at-2-27-27-pm.png

Oftentimes, definitions will be used based on the information provided from the ‘Given’ statement, as is the case here. Notice, at the end of the proof, that Substitution is used twice. Because each Substitution step is dealing with a different aspect of the geometric figure, we can categorize them as different steps. Keep in mind, as long as it makes sense, a proof can be as many steps as needed.

This next example involves angle measures:

Screen Shot 2018-02-20 at 2.40.28 PM.png

As you can see, with almost any geometric term, you can use its definition, even if not followed by a theorem or postulate. 

Finally, here is a proof involving Vertical Angles:

For this, you’ll notice that I didn’t use Definition of Vertical Angles. This is because, when an aspect of the figure is specifically told to you in the given statement, we can automatically assume it is true without proving it through the use of a definition.

Screen Shot 2018-02-20 at 2.52.02 PM.png

In addition, you’ll notice a new theorem: the Triangle Sum Theorem, which states that the interior angles of a triangle add up to 180 degrees. For my blog, it’s nearly impossible to explain every theorem out there that corresponds to the topic. However, as I move forward, I’ll continue to pull in relevant (and new) theorems, so that you can at least be introduced to them. As usual, there are so many ways of proving something true.

Over the next several months, we’ll continue to explore the complex structure and factors involved with completing proofs. Thank you so much for reading!

Gabe

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