


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 c – b < 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:

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, a^{2} + b^{2} = c^{2}:
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.
ExtraSpecial Right Triangles
Right triangles are pretty special in their own right.
But there are two extraspecial right triangles.
They are 306090 triangles and 454590 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 timesaving 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.
306090 Triangles
The guy who named 306090 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 306090 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 306090 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 306090 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 extraspecial
feature of 306090 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 306090 triangles.
Knowing how equilateral and 306090 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.
454590 Triangles
A 454590 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 306090 triangle,
the lengths of the sides of a 454590 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 454590 triangles.
It will save you time and may even save your butt.
Also, just as two 306090 triangles form
an equilateral triangles, two 454590 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 mathgeek 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:

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}
/_{3} AC.
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:
 All the corresponding sides of the two triangles are equal. This is known as the SideSideSide (SSS) method of determining congruency.
 The corresponding sides of each triangle are equal, and the mutual angles between those corresponding sides are also equal. This is known as the SideAngleSide (SAS) method of determining congruency.
 The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the AngleSideAngle (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 306090 triangle.
The hypotenuse of this 306090 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}/
_{2} bh =
^{1}/_{2} (9)(2) = 9.
