Types of Triangles at a Glance

What is the measure of ∠DFE? What sort of triangle is ∆DEF?

From the picture, we can see that the three sides of the triangle, DF, FE, and ED are all congruent and therefore have the same length. All the angles are congruent as well. That makes ∆DEF an equilateral triangle.

How can we figure out the angles of an equilateral triangle? Spoiler alert: They're all equal. We'll call each angle measurement x. Since they're all equal and we know all of a triangle's angles add up to 180°, we can set up this equation.

x + x + x = 3x = 180
x = 60

In other words, all the angles in an equilateral triangle have measurements of 60°, including our dear ∠DFE.

What is the measure of ∠BCD? What kind of triangle is ∆BCD? 

Since we know that ∠ABC is an exterior angle of ∆BCD, we can use the Exterior Angle Theorem to help us find ∠BCD. Remember the Exterior Angle Theorem?

m∠ABC = m∠BCD + m∠CDB

Well, we know that m∠ABC = 120, but what about ∠CDB? Luckily, its exterior angle, ∠CDF, is the same as ∠CDE + ∠EDF.

m∠CDB = 180 – (m∠CDE + m∠EDF)

Instead of substituting equations within equations (equation-ception?), we can solve them before we plug them back into the equation we had before.

We already know that the angles in an equilateral triangle are 60° in measure, so that takes care of m∠EDF. That means it's time for some number action.

m∠CDB = 180 – (64 + 60)
m∠CDB = 56

Now we can use that to find m∠BCD.

m∠ABC = m∠BCD + m∠CDB
120 = m∠BCD + 56
m∠BCD = 64

The remaining angle, ∠CBD, can be calculated using the Angle Sum Theorem or its 120° supplementary angle. Either way, we should get 60°.

That means our triangle has angles of 56°, 60°, and 64°. Three acute angles make an acute triangle, and that's what we have here. Also, since each of the sides has a different length, this bad boy is an acute scalene triangle.

Find the measure of ∠CHG. What kind of triangle is ∆CHG?

For this problem, we don't have exterior angles to help us out. All we know is that all the angles add up to 180°. We gotta start somewhere.

180 = m∠CHG + m∠HGC + m∠GCH

Luckily, m∠CHG = m∠HGC, so we can replace them both with another term. The letter x is too common, so let's use q. It's quacky and quivering with quirkiness.

180 = 2q + m∠GCH

What about ∠GCH? How are we supposed to figure out its measure? We should be careful not to guess too high a number. We don't want it to think we're calling it fat.

Actually, we can figure out its exact measure. Since ∠GCH and ∠BCD are vertical angles, they're congruent and have the same measure. That means m∠GCH = 64. We're safe, since 64° is cute and acute, much like saying, "Of course you aren't fat, honey. You're the perfect size."

180 = 2q + 64
q = 58

That means m∠CHG = 58.

What about the type of triangle? If ∠CHG and ∠HGC being congruent weren't enough, CH and CG are congruent as well. That's a clear-cut isosceles. Pair that up with its three acute angles, and ∆CHG is an acute isosceles triangle.