Hey there! If you’ve made it here to my third guide, you understand how lines work to form different types of angles, and you’re ready dive into the universe of the shapes they form. A triangle is the simplest shape with just three sides, so let’s start there. I’m old school and like paper notes, so my recommendation would be for you to print out the above tutorial to use as a guide while you’re working problems on your own or with friends. If you’re new school and you prefer digital resources, feel free to download and share. And remember, I’m always just a click away if you want to set up a consultation to figure out a schedule for math tutoring that works best for you.

I WAS OK WITH THE ANGLES. BUT TRIANGLES? CAN WE TALK?

Alright, let's start with the basics. Remember that question about what is the sum of angles in a triangle? The answer to this is the angle sum property, and it's 180 degrees. Now, let's turn up the heat and look at some different types of triangles.

## Equilateral Triangles:

The word equilateral kind of sounds like the word equal, right? Well, that what this triangle is all about. All sides have equal length, and all angles have equal measures of 60 degrees. Perfectly symmetrical and easy on the eyes, equilateral triangles are like the trendsetters of geometric perfection.

## Isosceles Triangles:

Sadly there’s no trick to remembering that Isosceles triangles have two equal sides. I mean, remembering the crazy spelling in challenging enough. (insert eyeroll emoji) Anyway, the angles that are opposite these two equal sides are also equal and form congruent base angles.

## Scalene Triangles:

Scalenes are a mess. They’re just all over the place. All three sides are different lengths, and no two are the same. Scalene triangles are the free spirits of geometry.

Do you notice a pattern here? If all sides are equal, then all angles are equal. If two sides are equal, then two angles are equal. If no sides are equal, then no angles are equal. Hmmmm……… looks like it’s time to roll your sleeves up and get ready to work.

Just like all the angles and parallel lines had relationships, and converse relationships in the last unit, you will see that sides and angles on separate triangles have relationships too. Getting a handle on congruence, was the name of the game with the line relationships, and this continues with triangles. Remember, congruency means that two shapes are exactly the same and there are a bunch of way to prove congruency in traingles including the Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) postulates as well as the Angle-Angle-Side (AAS), Hypoteneuse-Leg (HL), Hypoteneuse-Angle (HA), and Leg-Leg (L-L) theorems. Understanding what each of these mean and how to use them empowers you to establish congruency between separate triangles and unlock a deeper understanding of their interconnectedness.

As in all things Geometry, seeing and making constructions is needed to really thoroughly understand a concept. These postulates and theorems are no different. Each and every one of them ARE illustrated for you in the downloadable study guide, but I'll get you started on some ideas about the SAS Postulate and the AAS Theorem here.

## The SAS Postulate:

SAS stands for "side-angle-side," and the SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. In simpler terms, if you can match up a side, an angle, and another side of one triangle with the corresponding elements of another triangle, the two triangles are the same.

## The AAS Theorem:

AAS stands for "angle-angle-side" and the AAS theorem states that if two angles and one side in a first triangle are congruent to two angles and a side of a second triangle, then the two separate triangles are congruent and have the same size and shape. Once you have Angle-Angle-Side, you confidently declare that the two triangles are congruent. It's a bit like saying, "These triangles are a perfect match!"