


Triangles
The importance of triangles to the plane geometry questions
on the Math IC test cannot be overstated. Not only will you encounter
numerous questions specifically about triangles, you will also need
a solid understanding of triangles in order to answer other questions about
polygons, coordinate geometry, and trigonometry. Luckily for you,
the essential rules governing triangles are few and simple to master.
Basic Properties
Every triangle adheres to four main rules, outlined below:
1. Sum of the Interior Angles
If you were trapped on a desert island and had to take
the Math IC test, this is the one rule about triangles you should
bring along: the sum of the measures of the interior angles is 180º.
With this rule, if you know the measures of two of a triangle’s
angles, you will be able to find the third. Helpful, don’t you think?
2. Measure of an Exterior Angle
Another property of triangles is that the measure of an
exterior angle of a triangle is 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 (the point
at which two sides meet). An exterior angle is always supplementary
to the interior angle with which it shares a vertex, and equal in
measure to the sum of the measures of the remote interior angles.
Take a look at the figure below, in which d, the
exterior angle, is supplementary to interior angle c:
It doesn’t matter which side of a triangle you
extend to create an exterior angle; the exterior angle will always
be supplementary to the interior angle with which it shares a vertex
and therefore (because of the 180º rule) equal to the
sum of the remote interior angles.
3. Triangle Inequality
The third important property of triangles is the triangle
inequality rule, which states: the length of a side of a triangle
is 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.
Observe the figure below:
From the triangle inequality, we know 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.
If this angle is large (close to 180º) then a will
be large (close to b + c). If this angle is small
(close to 0º), then a will be small (close to b
– c).
For an example, take a look at this triangle:
Using the triangle inequality, we 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.
4. Proportionality of Triangles
This brings us to the last basic property of triangles,
which has to do with 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. Conversely, side c is the shortest side
and is the smallest
angle. It follows, therefore, that c < b < a and C <
B < A. This proportionality of side lengths and angle
measures holds true for all triangles.
We did not assign measures and lengths to the angles and
sides for the figure above. If we had limited information about
those values, however we could make certain assumptions about the
other unknown side lengths and angles measures. For example, if
we knew the measures of two of the angles in the triangle, we could
find the measure of the third angle and therefore decide which side
is the longest (it would be the side opposite the largest angle).
This is the kind of reasoning that you might have to use when dealing
with triangles on the test.

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 value 7 is the only choice
that fits the criteria.
Special Triangles
There are several special triangles that have particular
properties. Knowing these triangles and what makes each of them
special will help you immeasurably on the Math IC test.
But before getting into the different types of special
triangles, we must take a moment to explain the markings we use
to describe the properties of each particular triangle. For example,
the figure below has two pairs of sides of equal length and three
congruent angle pairs: these indicate that the sides have equal
length. The arcs drawn into and indicate
that these angles are congruent. In some diagrams, there might
be more than one pair of equal sides or congruent angles. In this
case, double hash marks or double arcs can be drawn into a pair
of sides or angles to indicate that they are equal to each other,
but not necessarily equal to the other pair of sides or angles:
Now, on to the special triangles.
Scalene Triangles
A scalene triangle has no equal sides and
no equal angles.
In fact, the special property of scalene triangles is
that they don’t really have any special qualities. Scalene triangles
almost never appear on the Math IC.
Isosceles Triangles
A triangle that contains two sides of equal length is
called an isosceles triangle. In an isosceles triangle,
the two angles opposite the sides of equal length are congruent.
These angles are usually referred to 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. For example,
if one base angle of an isosceles triangle is 35º, then you know that
the other base angle is also 35º. Since the three angles in a triangle
must add up to 180º, you can figure out the value of the third angle:
180º – 35º – 35º = 110º.
Equilateral Triangles
A triangle whose sides are all of equal length is called
an equilateral triangle. All three angles in an equilateral
triangle are congruent as well; the measure of each is 60º.
If you know that a triangle has three equal sides, then
the proportionality rule states that the triangle must also have
three equal angles. Similarly, if you know that a triangle has three
equal angles, then you know it 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 of the right triangle, and the other two sides are called
legs. The angles opposite the legs of a right triangle are complementary.
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.
The Pythagorean Theorem
The Pythagorean theorem is crucial to answering most of
the righttriangle questions that you’ll encounter on the Math IC.
The theorem will also come in handy later on, as you study coordinate
geometry and trigonometry. The theorem states that 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 theorem states that the square of the hypotenuse
is equal to the sum of the squares of the two legs.
If you know the measures of two sides of a right triangle,
you can always use the Pythagorean theorem to find the third.
Pythagorean Triples
Because right triangles obey the Pythagorean theorem,
only a few have side lengths which 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 Math IC is full of right triangles whose side lengths
are Pythagorean triples. Study the ones above and their multiples.
If you can recognize a Pythagorean triple on a triangle during the
test, you can drastically reduce the amount of time you need to
spend on the problem since you won’t need to do any calculations.
Special Right Triangles
Right triangles are pretty special in their own right.
But there are two extraspecial right triangles
that appear frequently on the Math IC. They are 306090 triangles
and 454590 triangles.
306090 Triangles
A 306090 triangle is a triangle with angles of 30º,
60º, and 90º. What makes it special is the specific pattern that
the lengths of the sides of a 306090 triangle follow. Suppose
the short leg, opposite the 30 degree angle, has length x.
Then the hypotenuse has length 2x, and the long
leg, opposite the 60 degree angle, has length x. The sides of every 306090 triangle will
follow this ratio of 1 : 2 :.
The constant ratio of the lengths of the sides of a 306090
triangle means that if you know the length of one side in the triangle,
you will immediately know 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 will know that the hypotenuse
is 4 meters long, and the leg opposite the 60º angle is 2 meters. On the Math IC, you will
quite often encounter a question that will present you with an unnamed
306090 triangle, allowing you to use your knowledge of this special
triangle. You could solve these questions by using the Pythagorean
theorem, but that method takes a lot longer than simply knowing
the proper 306090 ratio.
454590 Triangles
A 454590 triangle is a triangle with two 45º angles
and one right angle. This type of triangle is also sometimes referred
to as 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 that you should
know. If the legs are of length x (they are always
equal), then the hypotenuse has length x. Take a look at this diagram:
As with 306090 triangles, knowing the 1: 1: ratio for 454590 triangles can
save you a great deal of time on the Math IC.
Similar Triangles
Two triangles are called similar if the ratio of the lengths
of their corresponding sides is constant. In order for this to be
true, the corresponding angles of each triangle must be congruent.
In essence, similar triangles have exactly the same shape but not
necessarily the same size. Take a look at a few similar triangles:
As you may have assumed from the above figure, the symbol
for “is similar to” is ~. So if triangle ABC is
similar to triangle DEF, you will write ABC ~ DEF.
When you say that two triangles are similar, it is important
to know which sides of each triangle correspond to each other. After
all, the definition of similar triangles is that “the ratio of the
lengths of their corresponding sides is constant.” So, considering
that ABC ~ DEF, you know that
the ratio of the short sides equal the ratio of the larger sides.
^{AB}/_{DE} =
^{BC}/_{EF} =
^{CA}/_{FD} .
Just as similar triangles have corresponding sides, they
also have corresponding angles. If ABC ~ DEF,
then
Occasionally, the Math IC may present you with two separate
triangles and tell you that the two are similar. More often, the
Math IC will present 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 AED.
After presenting you with a diagram like the one above,
the Math IC will test whether you understand similarity by asking
a question like:

This question doesn’t tell you outright that DE and CB are
parallel, but it implicitly tells you that the two lines are parallel
by indicating that both lines form the same angle, xº, when
they intersect with BA. Once you realize that ABC ~ AED,
you know 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, you have to plug it into the proportion
along with CB:
Area of a Triangle
It’s quite likely that you will have to calculate the
area of a triangle for the Math IC. 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).
In the previous sentence we said “a base” 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
is the base until you designate one. The height of the triangle
depends on the base, which is why the area formula always works,
no matter which side you choose to be the base. The heights of a few
triangles are pictured with their altitudes drawn in as dotted lines.
Study the triangle on the right. Its altitude does not
lie in the interior of the triangle. This is why the altitude of
a triangle is defined as a line segment perpendicular to the
line containing the base and not simply as perpendicular
to the base. Sometimes the endpoint of the altitude does not lie
on the base; it can be outside of the triangle, as is the case of
the second example above.
On the Math IC, you may be tested on the area of a triangle
in a few different ways. You might be given the altitude of a triangle
along with the length of the base, but it’s unlikely you’d get such
an easy question. It’s more probable that the altitude would have
to be found, 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 the
short side is 2 and the medium side, which is also the altitude
of the original triangle, is 2. Now you can use the area formula to
find the area of the original triangle: ^{1}⁄_{2}bh = ^{1}⁄_{2}(9)(2) = 9 ≈ 15.6.
