# Congruent Triangles

### Topics

Since all triangles have angles, they can all be classified as acute, obtuse, or right. We just did that. On the other hand, triangles aren't made of *only* angles. They also have sides that we can classify. That means we can also distinguish triangles based on their relative side lengths.

For instance, an **equilateral triangle** is a triangle whose three sides are equal in length. Wanna construct one? We've gotcha covered.

We don't call these triangles "equilateral" because we want the other triangles to beat them up during lunch. (Honestly, who names their kid Equilateral, anyway?) We call them equilateral because all their side lengths equal each other, but also because all their angle measures equal each other (that's called being **equiangular**). We'll prove this a little later, but for now, you'll have to trust us.

A triangle that has two sides of equal length is called an **isosceles triangle**. We can get all sorts of isosceles triangles. There are acute isosceles triangles, obtuse isosceles triangles, and right isosceles triangles. They all have different angles, but since isosceles refers to sides only, they're all are equal in terms of their isosceles-ness.

If each side of a triangle goes its own way in terms of side length, we say that it is a **scalene triangle***.* Actually, that's not really true. They're called scalene, but we usually don't say anything about it. So many triangles are scalene that mathematicians just assume triangles are scalene unless stated otherwise. That's called being scalene-normative, but since we're studying math too, we'll succumb to peer pressure.

### Sample Problem

What would be the best description of ∆*ABC* if ∠*ACD* = 79°?

We can describe triangles by their angles and by their sides, and a good place to start is the triangle itself. That means we look at the picture.

We don't know about *AB*, but the tick marks on *AC* and *CB* mean that those two sides are equal in length. Two equal sides means isosceles, so what we have here is an isosceles triangle.

If we take a quick look at the angles, we can be even more specific in our description of ∆*ABC*. Since exterior angle ∠*ACD* is 79°, we know that interior angle ∠*ACB* has a measure of 180° – 79° = 101°. An interior angle over 90° is all we need for an obtuse triangle. That means ∆*ABC* is an obtuse isosceles triangle.

## Sample Problem

What would be the best description of ∆*KLM*?

To best describe this triangle, it's best to look at its angles and its sides. Each side of this triangle has a different number of tick marks. Like snowflakes or hipsters, each one is unique and different from the rest (well, maybe not hipsters). Even though we don't have to say it 'cause we're all scalene-normative, ∆*KLM* is scalene.

What about its angles? We know two of the triangle's exterior angles. We can get by with a little help from our friends, supplementary angles. Since we know ∠*JKL* = 123°, we also know that ∠*LKM* = 180° – 123° = 57°. One acute angle isn't enough to say anything conclusive, so we'll continue.

If ∠*LMN* = 153°, then its supplement, ∠*KML* equals 180° – 137° = 43°. Another acute angle. That means it all depends on the final angle.

Since we now know two of the three interior angles of the triangle, the Angle Sum Theorem for triangles allows us to find the last one.

m∠*LKM* + m∠*KLM* + m∠*KML* = 180

57 + m∠*KLM* + 43 = 180

m∠*KLM* = 80

So our triangle has angles of 43°, 57°, and 80°. All of 'em are under 90°, so they're all acute. Three acute angles translate to a for-sure acute triangle.