Triangles are closed figures containing three angles and three sides. The sum of the three angles in a triangle will always equal 180º. This is a very important fact. You must know it. There are two other important rules of triangles that you should know for the SAT.

There are a number of specialized types of triangles. Each of these types of triangles have special properties. The SAT will definitely test your understanding of these properties.

A scalene triangle has no equal sides and, therefore, no equal angles.

The special property of this triangle is that it doesn’t really have any special properties. SAT questions don’t usually deal with scalenes.

Isosceles triangles have two equal sides, in this case sides a and b (the little marks in those two sides mark the sides as being congruent or equal in length). The angles opposite the congruent sides are also equal, in this case the angles marked by xº and yº.

Because two of the angles of the isosceles triangle are equal and all triangles contain exactly 180º, if you know the value of one of the two equal angles, you can figure out the value of all the angles in the triangle. For example, if you know the value of you know the value of since and are equal. Angle z is equal to 180º – 2x (since x and y are equal, x + y = 2x). If you know the measure of you can figure out the measures of and since each equals ^{180 – x} /₂.

The SAT will test your knowledge of isosceles triangles. It might give you the length of a side and ask you the length of the other side to test your understanding of congruence. It might give you the value of an angle and ask you to figure out the value of another angle. It might ask you something else a little more indirect. But if you know these rules and remember them each time you see an isosceles triangle, you’ll do fine.

An equilateral triangle is a triangle in which all the sides and all angles are equal. Since the angles of a triangle must total 180º, the measure of each angle of an equilateral triangle must be 60º.

A triangle with a right angle (90º) is called a right triangle. Because the angles of a triangle must total 180º, the non-right angles ( and in the diagram below) in a right triangle must add up to 90º. The side opposite the right angle (side c in the diagram below) is called the hypotenuse.

There are many different types of right triangles, but two are particularly important for the SAT.

A 30-60-90 triangle is true to its name: it has angles of 30º, 60º, and 90º. A 30-60-90 triangle is actually half of an equilateral triangle. If you imagine an equilateral triangle and then cut it down the middle, you’ll end up with a 30-60-90 (knowing this fact can often help you on SAT problems).

As the diagram shows, the ratio between the three sides of a 30-60-90 triangle is always the same. The side opposite the 90º angle is always twice as long as the side opposite the 30º angle. The side opposite the 60º angle is always times as long as the side opposite the 30º angle. If you know these ratios and come across a 30-60-90 triangle during the SAT, you could spare yourself a lot of calculation. Note that these side lengths are ratios. A 30-60-90 triangle could have sides that measure 3, 6, and or 50, 100, and .

A 45-45-90 triangle lives a double life: it is both an isosceles triangle and a right triangle.

As the figure shows, the sides of this type of triangle always adhere to the same ratio. The side opposite the 90º angle is always times larger than the two equal sides that sit opposite the 45º angles.

The Pythagorean theorem defines the vital relationship between the sides of every right triangle (and that means every right triangle, not just the special ones we’ve already talked about). The theorem states that the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the legs.

If a triangle is a right triangle, this formula will always hold. Conversely, if the formula holds for a particular triangle, you know that triangle is a right triangle. If you are given any two sides of a right triangle, you can use this formula to calculate the length of the third side.

Certain groups of three integers can be the lengths of a right triangle. Such groups of integers are called Pythagorean triples. Some common Pythagorean triples include {3, 4, 5}, {5, 12, 13}, {8, 15, 17}, {7, 24, 25}, and {9, 40, 41}. Any multiple of one of these groups of numbers also can be a Pythagorean triple. For example, {9, 12, 15} = 3{3, 4, 5}. If you know these basic Pythagorean triples, they might help you quickly determine, without calculation, the length of a side of a right triangle in a problem that gave you the length of the other two sides.

In reference to triangles, the word similar means “of the same shape.” Two triangles are similar if their corresponding angles are equal. If this is the case, then the lengths of corresponding sides will be proportional to each other. For example, if and are similar, then sides AB and DE correspond to each other, as do BC and EF, and CA and FE.

That corresponding sides are proportional means that AB/DE = BC/EF = CA/FD.

Similarity can be very helpful on the SAT. For example, let’s say you come across the following question:

If you know the rule of similarity, then you can see that the ratio of CD:CA is 4:9, and know that CE:CB must obey the same ratio. Since EB is equal to 10, the only possible length of CE is 8, since 8:18 is equivalent to 4:9.

Trickier questions on the SAT might not tell you whether two triangles are similar. However, they will include information that will allow you to see that the two triangles are similar. If two pairs of corresponding angles are equal or if one pair of angles is equal and the two pairs of adjacent sides are proportional, then you know that two triangles are similar.

Congruence is another helpful rule of triangles. Congruence means that two triangles are identical. Some questions or images may state directly that the two triangles pictured are congruent. Some questions may include congruent triangles without explicit mention, however. Two triangles are congruent if they meet any of the following criteria:

The perimeter of a triangle is equal to the sum of the lengths of the triangle’s three sides. If a triangle has sides of lengths 4, 6, and 9, then its perimeter is 4 + 6 + 9 = 19.

The area of a triangle is equal to ¹/₂ the base of the triangle times the height: A = ¹/₂ bh. For example, given the following triangle,

in which b = 8 and h = 4:

the area equals ¹ /₂ bh = ¹/₂ The height of the triangle must be perpendicular to the base. You will almost definitely have to calculate the area of a triangle for the SAT. Know this formula.