Triangles
Triangles
Triangles pop up all over the Math section. There are questions specifically about triangles, questions that ask about triangles inscribed in polygons and circles, and questions about triangles in coordinate geometry.
Three Sides, Four Fundamental Properties
Every triangle, no matter how special, follows four main rules.
1. Sum of the Interior Angles
If you were trapped on a deserted island with tons of SAT questions about triangles, this is the one rule you’d need to know:
The sum of the interior angles of a triangle is 180°.
If you know the measures of two of a triangle’s angles, you’ll always be able to find the third by subtracting the sum of the first two from 180.
2. Measure of an Exterior Angle
The exterior angle of a triangle is always supplementary to the interior angle with which it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex. In the image below, d is the exterior angle.
Since d and c together form a straight angle, they are supplementary: . According to the first rule of triangles, the three angles of a triangle always add up to , so . Since and , d must equal a + b.
3. Triangle Inequality Rule
If triangles got together to write a declaration of independence, they’d have a tough time, since one of their defining rules would be this:
The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
There you have it: Triangles are unequal by definition.
Take a look at the figure below:
The triangle inequality rule says that cb < a < c + b. The exact length of side a depends on the measure of the angle created by sides b and c. Witness this triangle:
Using the triangle inequality rule, you can tell that 9 – 4 < x < 9 + 4, or 5 < x < 13. The exact value of x depends on the measure of the angle opposite side x. If this angle is large (close to ) then x will be large (close to 13). If this angle is small (close to ), then x will be small (close to 5).
The triangle inequality rule means that if you know the length of two sides of any triangle, you will always know the range of possible side lengths for the third side. On some SAT triangle questions, that’s all you’ll need.
4. Proportionality of Triangles
Here’s the final fundamental triangle property. This one explains the relationships between the angles of a triangle and the lengths of the triangle’s sides.
In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.
In this figure, side a is clearly the longest side and is the largest angle. Meanwhile, side c is the shortest side and is the smallest angle. So c < b < a and C < B < A. This proportionality of side lengths and angle measures holds true for all triangles.
See if you can use this rule to solve the question below:
What is one possible value of x if angle C < A < B?
(A) 1
(B) 6
(C) 7
(D) 10
(E) 15
According to the proportionality of triangles rule, the longest side of a triangle is opposite the largest angle. Likewise, the shortest side of a triangle is opposite the smallest angle. The largest angle in triangle ABC is , which is opposite the side of length 8. The smallest angle in triangle ABC is , which is opposite the side of length 6. This means that the third side, of length x, measures between 6 and 8 units in length. The only choice that fits the criteria is 7. C is the correct answer.
Special Triangles
Special triangles are “special” not because they get to follow fewer rules than other triangles but because they get to follow more. Each type of special triangle has its own special name: isosceles, equilateral, and right. Knowing the properties of each will help you tremendously, humongously, a lot, on the SAT.
But first we have to take a second to explain the markings we use to describe the properties of special triangles. The little arcs and ticks drawn in the figure below show that this triangle has two sides of equal length and three equal angle pairs. The sides that each have one tick through them are equal, as are the sides that each have two ticks through them. The angles with one little arc are equal to each other, the angles with two little arcs are equal to each other, and the angles with three little arcs are all equal to each other.
Isosceles Triangles
In ancient Greece, Isosceles was the god of triangles. His legs were of perfectly equal length and formed two opposing congruent angles when he stood up straight. Isosceles triangles share many of the same properties, naturally. An isosceles triangle has two sides of equal length, and those two sides are opposite congruent angles. These equal angles are usually called as base angles. In the isosceles triangle below, side a = b and :
If you know the value of one of the base angles in an isosceles triangle, you can figure out all the angles. Let’s say you’ve got an isosceles triangle with a base angle of 35º. Since you know isosceles triangles have two congruent base angles by definition, you know that the other base angle is also 35º. All three angles in a triangle must always add up to 180º, right? Correct. That means you can also figure out the value of the third angle: 180º – 35º – 35º = 110º.
Equilateral Triangles
An equilateral triangle has three equal sides and three congruent 60º angles.
Based on the proportionality rule, if a triangle has three equal sides, that triangle must also have three equal angles. Similarly, if you know that a triangle has three equal angles, then you know it also has three equal sides.
Right Triangles
A triangle that contains a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. The angles opposite the legs of a right triangle are complementary (they add up to 90º).
In the figure above, is the right angle (as indicated by the box drawn in the angle), side c is the hypotenuse, and sides a and b are the legs.
If triangles are an SAT favorite, then right triangles are SAT darlings. In other words, know these rules. And know the Pythagorean theorem.
The Pythagorean Theorem
The Greeks spent a lot of time reading, eating grapes, and riding around on donkeys. They also enjoyed the occasional mathematical epiphany. One day, Pythagoras discovered that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. “Eureka!” he said, and the SAT had a new topic to test.
Here’s the Pythagorean theorem: In a right triangle, a2 + b2 = c2:
where c is the length of the hypotenuse and a and b are the lengths of the two legs.
The Pythagorean theorem means that if you know the measures of two sides of a right triangle, you can always find the third. “Eureka!” you say.
Pythagorean Triples
Because right triangles obey the Pythagorean theorem, only a specific few have side lengths that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse of length = 5.83.
The few sets of three integers that do obey the Pythagorean theorem and can therefore be the lengths of the sides of a right triangle are called Pythagorean triples. Here are some common ones:
{3, 4, 5}
{5, 12, 13}
{7, 24, 25}
{8, 15, 17}
In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}.
The SAT is full of right triangles whose side lengths are Pythagorean triples. Study the ones above and their multiples. Identifying Pythagorean triples will help you cut the amount of time you spend doing calculations. In fact, you may not have to do any calculations if you get these down cold.
Extra-Special Right Triangles
Right triangles are pretty special in their own right. But there are two extra-special right triangles. They are 30-60-90 triangles and 45-45-90 triangles, and they appear all the time on the SAT.
In fact, knowing the rules of these two special triangles will open up all sorts of time-saving possibilities for you on the test. Very, very often, instead of having to work out the Pythagorean theorem, you’ll be able to apply the standard side ratios of either of these two types of triangles, cutting out all the time you need to spend calculating.
30-60-90 Triangles
The guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles have angles of , , and . What’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x. The sides of every 30-60-90 triangle will follow this ratio of 1: : 2 .
This constant ratio means that if you know the length of just one side in the triangle, you’ll immediately be able to calculate the lengths of all the sides. If, for example, you know that the side opposite the 30º angle is 2 meters long, then by using the 1: : 2 ratio, you can work out that the hypotenuse is 4 meters long, and the leg opposite the 60º angle is 2 meters.
And there’s another amazing thing about 30-60-90 triangles. Two of these triangles joined at the side opposite the 60º angle will form an equilateral triangle.
Here’s why you need to pay attention to this extra-special feature of 30-60-90 triangles. If you know the side length of an equilateral triangle, you can figure out the triangle’s height: Divide the side length by two and multiply it by . Similarly, if you drop a “perpendicular bisector” (this is the term the SAT uses) from any vertex of an equilateral triangle to the base on the far side, you’ll have cut that triangle into two 30-60-90 triangles.
Knowing how equilateral and 30-60-90 triangles relate is incredibly helpful on triangle, polygon, and even solids questions on the SAT. Quite often, you’ll be able to break down these large shapes into a number of special triangles, and then you can use the side ratios to figure out whatever you need to know.
45-45-90 Triangles
A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 30-60-90 triangle, the lengths of the sides of a 45-45-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x:
Know this 1: 1: ratio for 45-45-90 triangles. It will save you time and may even save your butt.
Also, just as two 30-60-90 triangles form an equilateral triangles, two 45-45-90 triangles form a square. We explain the colossal importance of this fact when we cover polygons a little later in this chapter.
Similar Triangles
Similar triangles have the same shape but not necessarily the same size. Or, if you prefer more math-geek jargon, two triangles are “similar” if the ratio of the lengths of their corresponding sides is constant (which you now know means that their corresponding angles must be congruent). Take a look at a few similar triangles:
As you may have assumed from the figure above, the symbol for “is similar to” is ~. So, if triangle ABC is similar to triangle DEF, we write ABC ~ DEF.
There are two crucial facts about similar triangles.
  • Corresponding angles of similar triangles are identical.
  • Corresponding sides of similar triangles are proportional.
For ABC ~ DEF, the corresponding angles are The corresponding sides are AB/DE = BC/EF = CA/FD.
The SAT usually tests similarity by presenting you with a single triangle that contains a line segment parallel to one base. This line segment creates a second, smaller, similar triangle. In the figure below, for example, line segment DE is parallel to CB, and triangle ABC is similar to triangle AE.
After presenting you with a diagram like the one above, the SAT will ask a question like this:
If = 6 and = , what is ?
Notice that this question doesn’t tell you outright that DE and CB are parallel. But it does tell you that both lines form the same angle, xº, when they intersect with BA, so you should be able to figure out that they’re parallel. And once you see that they’re parallel, you should immediately recognize that ABC ~ AED and that the corresponding sides of the two triangles are in constant proportion. The question tells you what this proportion is when it tells you that AD = 2 /3AC. To solve for DE, plug it into the proportion along with CB:
Congruent Triangles
Congruent triangles are identical. Some SAT questions may state directly that two triangles are congruent. Others may include congruent triangles without explicit mention, however.
Two triangles are congruent if they meet any of the following criteria:
  1. All the corresponding sides of the two triangles are equal. This is known as the Side-Side-Side (SSS) method of determining congruency.
  2. The corresponding sides of each triangle are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency
    .
  3. The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency
Perimeter of a Triangle
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. Easy. Done and done.
Area of a Triangle
The formula for the area of a triangle is
where b is the length of a base of the triangle, and h is height (also called the altitude). The heights of a few triangles are pictured below with their altitudes drawn in as dotted lines.
We said “a base” above instead of “the base” because you can actually use any of the three sides of the triangle as the base; a triangle has no particular side that has to be the base. You get to choose.
The SAT may test the area of a triangle in a few ways. It might just tell you the altitude and the length of the base, in which case you could just plug the numbers into the formula. But you probably won’t get such an easy question. It’s more likely that you’ll have to find the altitude, using other tools and techniques from plane geometry. For example, try to find the area of the triangle below:
To find the area of this triangle, draw in the altitude from the base (of length 9) to the opposite vertex. Notice that now you have two triangles, and one of them (the smaller one on the right) is a 30-60-90 triangle.
The hypotenuse of this 30-60-90 triangle is 4, so according to the ratio 1: : 2, the short side must be 2 and the medium side, which is also the altitude of the original triangle, is 2. Now you can plug the base and altitude into the formula to find the area of the original triangle: 1/ 2bh = 1/2(9)(2) = 9.
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