While there are many different kinds of triangles, there are some rules that are specific to all triangles, and we will start with these. Figure 1, below, is a generic triangle. If you draw different kinds of triangles, these rules will always hold true.
The angles of any triangle will ALWAYS add up to 180°. For the three internal angles of a triangle: x° + y° + z° = 180°.
The biggest side is ALWAYS opposite the biggest angle and the smallest side is ALWAYS opposite the smallest angle. In this case, we can therefore see x must be the smallest angle, because it is opposite side A, which is the smallest side.
Any side of a triangle will always be less than the sum of the other two sides but greater than the difference of the other two sides. In Figure 1, if we take side C, for example, we can say
(B – A) < C < (B + A).
If we draw an external line, as in Figure 2, the angle formed will always be equal to the sum of the other two angles in the triangle.
In this case, n = x + y. That is because x + y + z = 180 and n + z = 180. Therefore n must equal x + y. This property will be true for any triangle
Perimeter and Area
The perimeter of any figure is the distance around the outside of the figure, or the sum of the sides of that figure. The perimeter in Figure 1 is A + B + C.
The area of any figure is the amount of space that is inside that figure. Each figure has a different formula for finding its area.
To find the area of a triangle, we always need two elements: the base and the height.
The base of any triangle can be any of its sides. The height of the triangle is the perpendicular distance from the base to the opposite angle. Here are some examples of triangles with different bases and heights.
Every triangle has three bases and three corresponding heights. To find the area, use the following formula:
There are three important triangle types on the GRE.
An isosceles triangle has two equal sides and two equal angles. In the figure to the right: A = C, and sides AB = BC. This is in line with rule 2 above: if the two sides are equal, their opposite angles must be equal as well.
NOTE: The height of an isosceles triangle always bisects the triangle, creating two equal right triangles.
An equilateral triangle has three equal sides and three equal angles. Because a triangle’s angles must add up to 180°, each angle of an equilateral triangle is always 60°.
A right triangle is any triangle with a 90° angle. The two perpendicular sides are called legs and the side opposite the right angle is called the hypotenuse.
There is a constant relationship between the legs and the hypotenuse called the Pythagorean Theorem. The Pythagorean Theorem states that the square of the hypotenuse will equal the sum of the squares of the legs.
a2 + b2 = c2
Try using the Pythagorean Theorem yourself on a few triangles:
32 + 42 = c2
9 + 16 = c2
25 = c2
5 = c
122 + b2 = 132
144 + b2 = 169
b2 = 25
b = 5
42 + 82 = c2
16 + 64 = c2
Special Right Triangles based on Sides
There are several common right triangles on the GRE, and if you are familiar with them, you will be able to recognize them easily on the exam.
Note two important things. First, the largest side is ALWAYS the hypotenuse. If two legs of a triangle are 3 and 5, the hypotenuse WILL NOT be 4. Try it with the Pythagorean Theorem and you will see why. Second, these values are ratios. This means that multiples of these triangles are also special right triangles. The 3 – 4 – 5 is also the 6 – 8 – 10 and the 9 – 12 – 15. The 5 – 12 – 13 is also the 10 – 24 – 26.
Special Right Triangles based on Angles
There are also two special right triangles that we identify by the measurements of their angles. The first is the 45° – 45° – 90° triangle.
As you can see from the figure, the 45° – 45° – 90° triangle is an isosceles triangle, so all the rules we know for isosceles triangles apply here. There are two equal sides and two equal angles. For that reason, there is a constant relationship between the legs and the hypotenuse. Whatever the legs are, the hypotenuse is always that times .
The 30° – 60° – 90° right triangle works in the exact same way as the 45° – 45° – 90° triangle, but with different dimensions.
As you can see from the diagram, the dimensions of the 30° – 60° – 90° triangle are x - x - 2x. It is imperative that you memorize this triangle and these dimensions.
One easy way is to remember our rule from above – the smallest side is opposite the smallest angle and vice versa. Since x is the smallest, it is always opposite the 30° angle, and 2x, which is the largest, must always be opposite the 90° angle (meaning it is always the hypotenuse). Since x is between 1x and 2x ( it is approximately 1.7 - between 1 and 2), it is always opposite the middle angle, 60°.
An equilateral triangle may be divided into two 30° – 60° – 90° right triangles. The height of an equilateral triangle always bisects the triangle, creating two equal right triangles that are both 30-60-90 right triangles.
Because of this property, from the side of an equilateral triangle we may always figure out the height. That means we do not need to be given the height of an equilateral triangle to find the area.
Look at the example in the figure on the left. The sides are all length 10, but with the height drawn, the triangle has been bisected. The base is now cut in half, creating two 30° – 60° – 90° triangles, each with x = 5 and 2x = 10. The height of the triangle, then, must be x, or, in this case, 5. The area, , would be.
Note: Since the sides of an equilateral triangle are always proportional, there is a special
formula for its area. If the side of the triangle is s, the formula is. On the above equilateral triangle it would be 100 / 4 = 25 .
Keep in mind that Mr. GRE knows that you have memorized common triangle structures, whether they be 3-4-5 triangles or 45-45 right triangles. The GRE isn't supposed to be "beatable" by preparation, so Mr. GRE gets around this by setting traps designed to fool people who have studied from second-rate GRE prep books. As a result, if you see 3 and 4 as sides of a triangle DO NOT ASSUME that the hypotenuse is 5, because Mr. GRE isn't always so nice.
Triangles with the same angles are always proportional to each other. These triangles are called similar triangles because we can relate them to each other. In each set of similar triangles, the same sides opposite the same angles are proportional. There are three ways similar triangles can appear on the exam.
Figure 1: Two triangles, same angles. Notice that the second triangle’s sides are half the first triangle. We can make this comparison because they have the same angles.
Figure 2: One triangle, with a parallel line cutting through its middle. This figure is composed of two triangles. They each share angle y, and their bases are parallel. For that reason, they both have angle x and angle z, and the smaller, inner triangle is proportional to the larger triangle.
Figure 3: Two triangles, connected at one vertex, forming vertical angles, with parallel bases. In this figure, the two triangles are similar because they share angle z, and their bases are parallel. In this case, angles x and y are on opposite sides from one another (because of alternate interior angles).
Example 1 (easy)
For the triangle shown, find L.
The small box in the corner signifies
a right triangle. The ratio of the two
legs is 12/16 = 3/4. It is a 3-4-5 triangle (4 times the 3-4-5
triangle); consequently, its hypotenuse
L = 4 × 5 = 20.
Or we could have used the Pythagorean Theorem to obtain:
L2 = 122 + 162
L2 = 400
L = 20
Example 2 (easy)
Calculate the length L for the triangle shown.
This is a right triangle,
a 45°- 45°- 90° triangle.
The length of a leg of such a triangle is:
times the hypotenuse.
Example 3 (medium)
A given isosceles triangle has two equal angles of 30°. The side common to the 30° angles has a length of 4. How long are the equal sides?
A sketch of the triangle is always helpful. Let x be the unknown length. You know how to use the properties of a right triangle to solve for the sides of a triangle, so if you have to solve for the side of a different kind of triangle, you can use a right triangle within the given triangle. Can you see how one of the triangles we've just discussed could be helpful in solving the problem? By dividing the isosceles triangle into 2 right triangles, we get two 30° - 60° - 90° triangles.
The ratio of the side adjacent to the 30° angle and the hypotenuse is : 2. Hence,
Example 4 (hard)
A triangle has angles of 45° and 75°. The side opposite the 45° angle has a length of 6. What is the length of the side opposite the 75° angle?
Sketch the triangle. The remaining angle is 180 - (75 + 45) = 60°. Again, see if you can solve the problem by creating right triangles. Form two right triangles and label the unknowns x, y, z. The side adjacent to the 60° angle is 1/2 the hypotenuse.
Hence, y = 3. The side opposite the 60° angle is x = 3 (the triangle is 3 times as big as the base 30° - 60° - 90° triangle shown previously). Since the legs of a 45° - 45° - 90° triangle are equal, z = x = 3. The length is then