6.1 Lines and Angles
6.2 Triangles
6.3 Polygons
6.4 Circles
6.5 Key Formulas
6.6 Review Questions
6.7 Explanations
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.
What is one possible value of x if angle C < A < B?
(A) 1
(B) 6
(C) 7
(D) 10
(D) 15
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 right-triangle 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 a2 + b2 = c2:
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 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 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 30-60-90 triangle will follow this ratio of 1 : 2 :.
The constant ratio of 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 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 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.
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 sometimes referred to 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 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 30-60-90 triangles, knowing the 1: 1: ratio for 45-45-90 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:
If = 6 and = , what is ?
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 = 23 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 30-60-90 triangle.
The hypotenuse of this 30-60-90 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: 12bh = 12(9)(2) = 9 ≈ 15.6.
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