Isosceles: A triangle with at least two equal angles

An isosceles triangle normally has two angles that are equal size and one angle whose size is unique. For example, a triangle with angles of 45°, 45° and 90° would count as isosceles since it has two equal angles (45° and 45°) and then a third unique angle (an "odd man out," 90°).

Since LMN is isosceles and we are given that it has a 60° angle, we know that this 60° angle must either be one of the two equal angles or the one unique angle (the "odd man out"). Think it through each way:

First, if the 60° angle we are given is one of the two equal angles, then it has another 60° angle as its pair:

Angle 1: 60°
Angle 2: 60°
Angle 3: ?

How many degrees does that leave for the third angle in the triangle? We don't know, so we can just call it x:

Angle 3: x

Now we can use algebra to solve for x. The three angles in a triangle always add to 180°. So the three angles we've described (the two 60° angles and the unknown angle x) must add up to that:

60° + 60° + x = 180°

So:

120° + x = 180°
x = 180° - 120°
x = 60°

Therefore, if the 60° angle we are given is one of the two equal angles, then the last angle turns out to be 60° as well!

Second, now let's try the other possibility. Above, we assumed the 60° angle we are given was one of the two paired angles in the isosceles triangle. But what if, instead, the 60° angle is the one unique angle (the "odd man out") and it is the other two angles which are the pair? In that case, the other two angles are equal to each other.

Angle 1: 60°
Angle 2 and 3: ??

How big are these other two equal angles? We don't know, so we use a new variable to figure it out:

Angle 2: y
Angle 3: y

Notice I've given the same variable to each of the two angles. This is key. We must do this because the two angles are equal. They get the same variable because they are the same thing.

The three angles in a triangle add up to 180°:

60° + y + y = 180°

So:

60° + 2y = 180°
2y = 180° - 60°
2y = 120°
y = 60°

We have found that if the 60° angle we were given is the "odd man out," then each of the other two angles also has a measure of 60°!

Therefore, on either assumption -- whether we start by letting the 60° angle we are given be one of the two equal angles or we start by letting it be the "odd man out" -- either way we end up with the same conclusion: all three angles are each equal to 60°.

Can this be right? We were told that the triangle is isosceles, but now we've determined that all three of the angles--not just two of them--are equal to one other. A triangle with all three angles equal is called an equilateral triangle. So why have we concluded that this triangle is equilateral, when we were plainly given that it is isosceles?

Here's the reason: An isosceles triangle is one in which at least two sides are equal. So it can still be isosceles if all three sides are equal. (Three is at least two.) Therefore, the triangle in this problem is both equilateral and isosceles. It does not have to just be one or the other. In fact, every equilateral triangle is an isosceles, since every equilateral triangle has at least two equal angles.

Where does this leave us? Here's another rule: In an equilateral triangle, not only are all the angles equal, but all the sides are equal as well. Therefore, C is the correct choice: The two sides of this triangle listed as Quantity A and Quantity B are equal.

Key Points:

1) An isosceles triangle is a triangle with at least two equal sides or angles.
2) An equilateral triangle is a triangle with all three angles equal.
3) Every equilateral triangle also counts as isosceles (since it has at least two equal angles).
4) Every equilateral has all three sides and all three angles equal.
5) The three angles in any triangle add up to 180°.
6) Sometimes, to solve a problem, you have two identify two possibilities and test the answer both ways.