I managed to get through the unit Congruence this week but unfortunately didn’t get much further than that. Though I’m a disappointed with my lack of progress this week, I spent close to 7 hours working through KA which makes me feel better about it. The unit Congruence had a number of videos I needed to watch that proved different types of congruency within shapes, and these videos took up a lot of time and was the main reason I didn’t get further into the following unit Similarity.
First off, one thing I took away from the unit Congruence is that shapes are dope. (I literally said that out loud when going through an exercise and made a note to write it in this post.) Congruence in geometric terms means “same size and shape”. When lines, angles, shapes, etc. are congruent, it means they literally have the exact same dimensions. I learned that the word “similar” in geometry means “shapes that have the same corresponding angles but don’t have the same lengths of sides”.
Sal made a point of defining the words “axiom” and “postulate” which I learned are two words that both essentially mean “a universal truth that cannot be proved but is taken for granted as being true”. Postulate in particular means “a theory based on a universal truth that can’t be proven”. Sal then talked about Congruent Triangle axioms/postulates. These axioms/postulates state that if two triangles both have specific combination of sides/angles, they will be congruent. The different combinations of side/angle axioms/postulates are:
- Side, Side, Side (S.S.S.)
- Side, Angle, Side (S.A.S.)
- Angle, Side, Angle (A.S.A.)
- Angle, Angle, Side (A.A.S.)
I found the easiest way for me to remember these axioms/postulates is by remembering and first writing out where the “S”s go and then filling in the “A”s:
- S. S. S.
- S. _ S.
- _ S. _
- _ _ S.
There’s also one more postulate that works but only if the angle is a right angle or obtuse angle (i.e. >90 degrees). That postulate is S. S. A. If the angle is not a right or obtuse angle however, the triangles could have two variations and therefore may not be congruent.
I was then reintroduced to Isosceles triangles and Equilateral triangles:
- Isosceles Triangles
- Two of the sides are congruent.
- The congruent sides are referred to as the “legs” and the non-congruent side is referred to as the base.
- The two angles that are apart of the base side are referred to as the “base angles”.
- Equilateral Triangles
- All sides of the triangle are equal.
- All three angles are always 60 degrees.
I briefly got into the next unit Similarity and learned about three Similar Triangle axioms. They are:
- A.A.
- K.(S.S.S.)
- K. (S.A.S.)
I’m still not completely sure how/why these axioms work, but as far as I know the K. in these axioms stands for the “proportion/ratio” between sides, i.e. if three sides in one triangle are 2, 3, and 4 units in length and three sides in another triangle are 4, 6, and 8 units in length, the triangles are similar and K = 2.
This coming week I’d like to get through all of the unit Similarity (480/700 M.P.) and get though a good chunk of the following unit Right Triangles and Trigonometry (100/1100 M.P.). When beginning KA, trigonometry was one of the subjects I was most keen to learn about only because it sounds like a difficult subject and I want to be able to say “I understand trigonometry” because I think it would be cool. My guess is this unit won’t get too deep into the subject since there is an entire course dedicated to Trigonometry but regardless, after 5 months of working on KA, I’m excited to finally be getting into it!
**Auston Powers meme** Who gets excited about trigonometry? Honestly.