6.1 Lines and Angles
6.2 Triangles
6.3 Polygons
6.4 Circles
6.5 Key Formulas
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 bc).
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 sp-ecial 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 a2 + b2 = c2, 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 extra-special right triangles that appear frequently on the Math IIC. They are 30-60-90 triangles and 45-45-90 triangles.
30-60-90 Triangles
A 30-60-90 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 30-60-90 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 30-60-90 triangle will follow this 1 : 2 : ratio.
The constant ratio in the lengths of the sides of a 30-60-90 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 30-60-90 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 30-60-90 ratio. The key is to be aware that there are 30-60-90 triangles lurking out there and to strike when you see one.
45-45-90 Triangles
A 45-45-90 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 30-60-90 triangle, the lengths of the sides of a 45-45-90 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 30-60-90 triangles, knowing the ratio for 45-45-90 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 right-most triangle in the figure above.
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