Unit 3 - 1
Key Objectives (from the notes):
- Apply theorems about the interior and exterior angles of triangles.
- Find the measure of interior and exterior angles of triangles.
- Prove theorems about isosceles and equilateral triangles.
- Apply properties of isosceles and equilateral triangles.
Here we are! Unit 3! I was rather excited making it to this point, already being two months into the engaging topic that is geometry. But nonetheless, I have a serious job here on this website to attend to...
The first thing we learned (learned - no, it was more like a review) in this lesson was the Triangle Angle Sum Theorem. I'll let this picture express what that means first:
The first thing we learned (learned - no, it was more like a review) in this lesson was the Triangle Angle Sum Theorem. I'll let this picture express what that means first:
But the rubric says that I absolutely HAVE to explain it myself. Well, alright. Basically what the content in the green box is trying to say is that if you take any triangle and add up the measure of each angle, the result will always, 100% of the time, without any doubt or question, be 180 degrees.
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We also "learned" corollaries and applications of this, and learned how to properly use the theorem in proofs. This builds off of what we learned these past couple of units, as pertaining to proofs. This also builds on the topic we've "learned" for the past four of five grade levels, if what we were taught then wasn't enough.
Second on the list is exterior angles. This points directly to the theorem, aptly named "Exterior Angle Theorem". Once again, let's let the picture have its say, first.
Second on the list is exterior angles. This points directly to the theorem, aptly named "Exterior Angle Theorem". Once again, let's let the picture have its say, first.
Yes, I know, rubric. Stop yelling at me for two seconds and let me say what I need to say. The... stuff... you see in the green, rounded rectangle to the left is the Exterior Angle Theorem, basically saying that if you take two angles in a triangle (in this case, 1 and 2), then the sum of the measures of said angles is equal to the measure of the exterior angle on the opposite side of the triangle (that's 4 in the picture).
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For some of us, that topic might have been new. I say "some of us" because I'm not certain that the mathematical proficiency of some of the other students were... as highlighted... as mine upon entering this class (no offense to the students I'm referring to right now - you just didn't know it beforehand). This theorem was simply a review for me, but as I've said, maybe not for others. Later, lessons will build upon this, so make sure you know it well.
A third thing that we learned was, well, the Third Angles Theorem (get it?). And as usual, take it away, pictorial.
A third thing that we learned was, well, the Third Angles Theorem (get it?). And as usual, take it away, pictorial.
What that material is trying to express is that if you already know two corresponding, congruent angles in two triangles, the remaining corresponding pair has to be congruent. If you think about it, it makes sense. I could write a proof for this theorem on this website, but that would be a pretty major digression, since we didn't learn that (ever notice how math class teaches you the content but doesn't always explain why the content is the way it is?).
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Well, there's not much else I can comment about this theorem. It's fairly self-explanatory. All I can think of to say is that this theorem will come in handy to save many, many tedious steps in upcoming proofs.
Next on the list is stuff about isosceles triangles and extensions of that onto equilateral triangles. First off, you should know what isosceles triangles are in case you don't. Isosceles triangles are triangles that have two congruent sides and two congruent angles. "Why both?" you might be asking. "Can't there ever be two congruent sides and no congruent angles? Or vice versa?" To answer your question, there's the Isosceles Triangle Theorem and its converse, as well as the corollaries onto equilateral triangles. I would show you a picture, but I kind of want the rubric to stop screaming at me every time I do, as if I don't get the point.
The Isosceles Triangle Theorem basically states that if two sides of a triangle are congruent, then the two base angles are also congruent. The converse of this (refer back to Unit 1 if you don't remember what a converse is) is also true. What this means is that if you have two congruent sides, there cannot be no congruent angles - those two base angles have to be congruent! And vice versa! Does that answer the question that you likely never even asked in the first place? I know, it does.
And the corollaries. Basically, if the Isosceles Triangle Theorem is true, then it supports the claim that any equilateral triangle is equiangular, and if the converse of the Isosceles Triangle Theorem is true, then it supports the claim that any equiangular triangle is equilateral. These theorems apply in great ways in coming lessons.
That about covers it for this lesson.
Next on the list is stuff about isosceles triangles and extensions of that onto equilateral triangles. First off, you should know what isosceles triangles are in case you don't. Isosceles triangles are triangles that have two congruent sides and two congruent angles. "Why both?" you might be asking. "Can't there ever be two congruent sides and no congruent angles? Or vice versa?" To answer your question, there's the Isosceles Triangle Theorem and its converse, as well as the corollaries onto equilateral triangles. I would show you a picture, but I kind of want the rubric to stop screaming at me every time I do, as if I don't get the point.
The Isosceles Triangle Theorem basically states that if two sides of a triangle are congruent, then the two base angles are also congruent. The converse of this (refer back to Unit 1 if you don't remember what a converse is) is also true. What this means is that if you have two congruent sides, there cannot be no congruent angles - those two base angles have to be congruent! And vice versa! Does that answer the question that you likely never even asked in the first place? I know, it does.
And the corollaries. Basically, if the Isosceles Triangle Theorem is true, then it supports the claim that any equilateral triangle is equiangular, and if the converse of the Isosceles Triangle Theorem is true, then it supports the claim that any equiangular triangle is equilateral. These theorems apply in great ways in coming lessons.
That about covers it for this lesson.