NOT MANDATORY! DO ONLY IF YOU CHOOSE TO!

 

The Triangle Inequality Theorem states:
"The sum of the lengths of any two sides of a triangle is greater than the length of the third side."A corollary to the theorem (an add-on if you will) is that the perpendicular segment from a point to a line is the shortest segment from the point to the line.Your task:
Create a short (30-60 second) video detailing this theorem and its corollary. The video is not you just reciting the theorem, or you drawing it out. It is showing the actuality of the rule. (hint-- think streets and sidewalks and the such, you know, lines that are real.)
Have a narrative of some sort to your video, again, don't just define it. Live it.

Interior Angles of a Regular Polygon

As we've seen with the (n-2)180 formula, the sum total of the measures of the interior angles of a polygon can be quickly ascertained. Further, each individual measure within can also be ascertained by dividing the sum total by the actual number of sides (n), as long as the polygon is regular.

As per the image above, the more sides of a regular polygon, the closer the shape comes to being a circle. With that, the interior angles are ever increasing towards a max possible value. So if instead of a decagon drawn inside that circle, I drew a 50-sided shape, it would be near impossible to see the shape inside the circle.

So three questions:

1) What is that max value the interior angle measures are approaching as you increase the number of sides?

2) Give an example of the measure of each interior angle of a (you choose)-sided regular polygon. 

3) What is the sum of the measures of the exterior angles of the polygon you've chosen?

My example for #2:

A 45-sided regular polygon has interior angle measure 172 degrees, because [(n-2)*180]/n gives 172 in my case. 

*You must choose a number that no other student has previously chosen and put into the comment section; shouldn't be a problem, as there are an infinite amount of numbers to choose from, so make sure by looking at the other examples. In addition, the number you choose must be greater than any named or previously discussed polygons. Also, if you can manage to choose a number that gives a whole number value (non-decimal) as the interior angle measure, bonus. First come, first serve with that.*

Contrapositive of a conditional statement

A conditional statement is one which has a hypothesis and a conclusion in the if-then format. So, for example, "If an angle is 45 degrees, then the angle is acute."

The converse of a conditional statement is where you switch the hypothesis and conclusion literally. From the same example, "If the angle is acute, then the angle measures 45 degrees."

Obviously, the converse is a false statement; an acute angle can measure 30 degrees, doesn't have to be 45.

A statement whose converse and original statement are both true statements is a biconditional statement.

My Example: If it is December 25th, then it is Christmas Day.

If it is Christmas Day, then it is December 25th.

Your turn!! (be careful. Make sure it is biconditional!)

Deductive and Inductive Reasoning

Okay. Now we're dealing with proofs. The main idea behind a proof is that you are defending an argument through sound reasoning. Watch this video detailing the two types of reasoning we engage in when trying to evidence a proof. The video explains inductive and deductive reasoning. 

Your Task:

1) Give an example of deductive reasoning with at least 3 steps (different, obviously from the video's example).

Here's mine: 

a) All crows are black 

b) The bird on my fence is a crow

c) Therefore, the bird on my fence is black

2) Then give an errant example of deductive reasoning, where there is a false conclusion.

Here's mine:

a) All Toyota cars are relatively fuel efficient

b) My car is relatively fuel efficient

c) Therefore, I drive a Toyota

Clearly, my reasoning is incorrect; I could drive a different relatively fuel efficient vehicle.

Now you go. The grade on this post is simple:

10 total points. 8 for actually posting, and 2 for commenting on another student's post as to whether you agree with his or her examples being deductive reasoning, and the errant example being indeed errant.

Finding geometry

The construction of your house is dependent on geometric truths. The walls, baseboards, floorboards, carpet, windows, roof, all of it. Your task, due by Monday September 10th, is to identify a unique (in your mind) or interesting geometric feature about your house, interior or exterior, photograph it if possible, post the photo here, and describe in detail the geometry you see.
Here's my entry:

This is a photo of my home under construction. It seems the roof, unlike say a treehouse, has a 'pitch', measuring 60 degrees from the ground my feet are on in the photo, which of course is parallel to the ground. I've wondered in the past, "Why 60 degrees? Why not 80? I guess 20 would be too close to the ground in most spots." I also notice the studs (the 2X4s running vertically next to the window) are all parallel, and all perfectly perpendicular with the floor.


Your turn!