The GMAT quantitative section asks, among other things, about geometry. One of the GMAT’s favorite figures is the **isosceles** triangle. An isosceles triangle is one that has two congruent sides. Knowing simply that about a triangle has profound implications for answer GMAT Problem Solving & Data Sufficiency questions.

## Euclid’s Remarkable Theorem

Euclid first proved this theorem over 2200 years ago. The theorem says:

**If the two sides are equal, then the opposite angles are equal**

and

**if the two angles are equal, then the opposite sides are equal.**

It can be used for deductions in both directions (logically, this is called a “biconditional” statement”). Another way to say this: a statement about equal angles is sufficient to conclude equal sides; conversely, a statement about equal sides is sufficient to conclude equal angles. Remember that on GMAT Data Sufficiency!

## Isosceles Triangles and the 180º Triangle Theorem

Another favorite GMAT geometry fact is that the sum of all three angles in a triangle − any triangle − is 180º. This is particularly fruitful if combined with the Isosceles Triangle Theorem.

Suppose you are told that Triangle ABC is isosceles, and one of the bottom equal angles (called a “base angle”) is 50º. Then immediately you know that the measure of the other base angle is also 50º, and that means the top angle (the “vertex angle”) must be 80º. Knowing the measure of one base angle is sufficient to find the measures of all three angles of an isosceles triangle.

Suppose you are told that Triangle ABC is isosceles, and the vertex angle is 50º. Well, you don’t know the measures of the base angle, but you know they’re equal. Let x be the degrees of the base angle; then . So, each base angle is 65º. Knowing the measure of the vertex angle is sufficient to find the measures of all three angles of an isosceles triangle.

BUT, if you are told that Triangle ABC is isosceles, and one of angles is 50º, but you don’t know whether that 50º is a base angle or a vertex angle, then you cannot conclude anything about the other angles in the isosceles triangle without more information. That’s a subtle but important distinction to remember on GMAT Data Sufficiency.

## Free Practice Questions on Triangles

http://gmat.magoosh.com/questions/1019

http://gmat.magoosh.com/questions/1024

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Question regarding an excerpt from your blog “Suppose you are told that Triangle ABC is isosceles, and the vertex angle is 50º. Well, you don’t know the measures of the base angle, but you know they’re equal. Let x be the degrees of the base angle; then x + x + 50º = 180º right 2x = 130º right x = 65º. So, each base angle is 65º. Knowing the measure of the vertex angle is sufficient to find the measures of all three angles of an isosceles triangle.”

My question is: How can we assume that the angles opposite to the vertex angles are the base angles? Since, as per the rule, any 2 angles can be equal, any one of the other two angles could either be equal to 50 or not. But we can’t be assume that the vertex angle is the different angle while the other two are equal. Very confused about this statement “Knowing the measure of the vertex angle is sufficient to find the measures of all three angles of an isosceles triangle.”

Hi Arushi,

You can only call the unique angle the “vertex angle” and only the identical angles are the “base angles.” So the nomenclature here tells you exactly how to approach this. 🙂

I had a question about an isosceles question I encountered on a GMAT prep.

The prompt: In isosceles triangle RST, what is the measure of angle R?

1. Angle T=100 degrees

2. Angle S= 40 degrees

Is the given info of RST enough to draw a rough estimate of the triangle and conclude that S is the vertex ? And if that is the case than angle T would be a base angle and equivalent to angle R. The answer to this question is statement 1 is sufficient.

Hi Holly,

The sum of the angle measurements of a triangle is 180˚. And one characteristic of isosceles triangles is that at least 2 of the 3 angles are equal. So, given (A), that angle T = 100˚, we know that angle T must be the largest angle and that the measure of angle S is equal to the measure of angle R. Otherwise, either R or S would equal T and the total angle measurement would be more than 180˚. So, given (A) we can determine the measure of angle R 🙂

angle R = angle S

angle T = 100

angle T + angle R + angle S = 180

100 + 2*(angle R) = 180

2*(angle R) = 80

angle R = 40On the other hand, we cannot determine which two angles are equal given only (B). That’s because either the measure of T or R could be equal to the measure of S. For that reason, (B) is not sufficient, and the answer is A.

Hope this helps 🙂