


Triangles
You will 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 easy to master.
Basic Properties
There are four main rules of triangles:
1. Sum of the Interior Angles
If you were stranded on a desert island and had to take
the Math IIC test, this is the one rule about triangles you should
bring along: the sum of the measures of the interior angles is 180º.
Now, if you know the measures of two of a triangle’s angles, you
will be able to find the third. Helpful rule, don’t you think?
2. Measure of an Exterior Angle
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.
Special Triangles
There are several special triangles that have particular
properties. Knowing these triangles and what makes each of them
special can save you time and effort.
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 A and B 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 properties. Scalene triangles
almost never appear on the Math IIC.
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 A=B.
There is no such thing as a triangle with two equal sides
and no congruent angles, or vice versa. From the proportionality
rule, if a triangle has two equal sides, then the two angles opposite
those sides are congruent, and if a triangle has two congruent angles,
then the two sides opposite those angles are equal.
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.
As is the case with isosceles triangles, if you know that
a triangle has either three equal sides or three congruent angles,
then you know that the other must also be true.
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, C 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 vital to most of the problems
on right triangles. It 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, a and b are the
lengths of the two legs, and the square of the hypotenuse is equal
to the sum of the squares of the two legs.
Pythagorean Triples
Because right triangles obey the Pythagorean theorem,
only a 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 triples:
{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’s a multiple of {3, 4, 5}.
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 IIC. 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 30º angle, has length x.
Then the hypotenuse has length 2x, and the long leg,
opposite the 60º degree angle, has length x. The sides of every 306090 triangle
will follow this 1 : 2 : ratio.
The constant ratio in the lengths of the sides of a 306090
triangle means that if you know the length of one side in the triangle,
you 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 know
that the hypotenuse is 4 meters long and the leg opposite the 60º
angle is 2 meters. On the
Math IIC 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. The key is to be
aware that there are 306090 triangles lurking out there and to
strike when you see one.
454590 Triangles
A 454590 triangle is a triangle with two 45º angles
and one right angle. This type of triangle is also known 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 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 ratio for 454590
triangles can save you a great deal of time on the Math IIC.
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 could 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 equals the ratio of the larger sides.
^{AB}/_{DE} =
^{BC}/_{EF} =
^{CA}/_{FD} .
Just as similar triangles have corresponding sides, they
also have congruent angles. If ABC ~ DEF,
then_{ }A = _{}D, _{}B = _{}E,
and _{}C = _{}F.
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 the
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. The measure of its height
does not lie in the interior of the triangle. The height of a triangle
is defined as a line segment perpendicular to the line containing
the base, and not just the base. Sometimes the endpoint
of the height does not lie on the base; it can be outside of the
triangle, as is the case in the rightmost triangle in the figure
above.
