Today, I’m going to begin jumping ahead and covering several main portions of Geometry, especially as my in class learning is about 7 chapters ahead. As I’ve always said about proofs, there are ‘infinite’ methods that can be used to achieve a desired solution. What I find most useful is proving triangle congruence. Once you prove triangles congruent, you can prove so many different relationships, like congruent segments, angles, and even aspects like corresponding arcs when we begin talking about circles. No matter what, this is a tactic you’ll want to know.
Proving two triangles congruent is really quite simple, and there are five separate methods that you can use depending on the information you are given.
Let’s get started.
Side-Side-Side (SSS) Congruence Postulate
This theorem states that if two separate triangles have corresponding side lengths that are all congruent, then the two triangles are congruent.
I think here it’s also important to note how we name triangles. Let’s say we had triangles ABC and EFG, with each vertices being assigned a letter, listed in order of rotation (it can either be clockwise or counterclockwise, just the sequence of the letters matter- ie, the same triangles could be triangles CBA and GFE and it wouldn’t make a difference.
Another thing to note is that when two letters are listed sequentially, this indicates a segment (ie in the same triangles, segments AB, BC, CA, EF, FG, and GE). In triangles, any combination of sequential vertices creates a segment. And an easy way to determine which sides are corresponding are that, in order of the vertices listed, the first two letters of each triangle are corresponding, the last two, and the first and the last. This is also true with any polygon, just with greater combinations.
Therefore, if you note that a triangle is congruent because all three sides are congruent, you must name them properly in a congruence statement based on (especially if the triangle is scalene or isosceles) which dimensions match up.
A congruence statement looks like this:
(Triangle)ABC ≅ (Triangle) EFG
I’ll give you a visual diagram below demonstrating each theorem and postulate, but first, here are the rest of them:
Side-Angle-Side (SAS) Congruence Postulate
When you are given two segment lengths with an angle measure directly in between them that is congruent to the corresponding measures within a second triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA) Congruence Postulate
When you are given two angle measures with one segment length directly in between them that is congruent to the corresponding measures within a second triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS) Congruence Theorem
When you are given two consecutive angle measures followed by a side length touching one of the angle measures, all of which are congruent to the corresponding measures within a second triangle, then the two triangles are congruent.
Hypoteneuse-Leg (HL) Congruence Theorem
This can only be used with a right triangle, but in two right triangles, if the hypotenuses (the longest sides opposite the right angles) and one corresponding leg is congruent to their corresponding measures in the second triangle, then the two triangles are congruent.
Hopefully this will be of help to you as a base method when solving future proofs, which we’ll get more in depth with soon.
Thank you so much for reading!