Angles in Triangles
photo taken by Joe Sueyoshi
A student will be able to:
The word trigonometry derives from two Greek words meaning triangle and measure. As you will learn throughout this chapter, trigonometry involves the measurement of angles, both in triangles, and in rotation (e.g, like the hands of a clock.) Given the important of angles in the study of trigonometry, in this lesson we will review some important aspects of triangles and their angles. We'll begin by categorizing different kinds of triangles.
Video: Angles and types of triangles
Formally, a triangle is defined as a sided polygon. This means that a triangle has sides, all of which are (straight) line segments. We can categorize triangles either by their sides, or by their angles. The table below summarizes the different types of triangles.
|Equilateral/equi-angular||A triangle with three equal sides and congruent angles||This type of triangle is acute.|
|Isosceles||A triangle with equal sides and two equal angles||An equilateral triangle is also isosceles.|
|Scalene||A triangle with no pairs of equal sides|
|Right||A triangle with one angle||It is not possible for a triangle to have more than one angle (see below.)|
|Acute||A triangle in which all angles measure less than|
|Obtuse||A triangle in which one angle is greater than||It is not possible for a triangle to have more than one obtuse angle (see below.)|
In the following example, we will categorize specific triangles.
Example 1: Determine which category best describes the triangle:
a. A triangle with side lengths and
b. A triangle with side lengths and
c. A triangle with side lengths and
a. This is a scalene triangle.
b. This is an equilateral, or equiangular triangle. It is also acute.
c. This is a scalene triangle, but it is also a right triangle.
While there are different types of triangles, all triangles have one thing in common: the sum of the interior angles in a triangle is always . You can see why this true if you remember that a straight line forms a "straight angle," which measures . Now consider the diagram below, which shows the triangle , and a line drawn through vertex , parallel to side . Below the figure is a proof of the triangle angle sum.
We can use this result to determine the measure of the angles of a triangle. In particular, if we know the measures of two angles, we can always find the third.
Animation: The sum of the interior angles of a triangle is 180 degrees
Example 3: Find the measures of the missing angles.
a. A triangle has two angles that measures and .
b. A right triangle has an angle that measures .
c. An isosceles triangle has an angle that measures .
The triangle is a right triangle, which means that one angle measures .
So we have: .
c. and , or and
There are two possibilities. First, if a second angle measures , then the third angle measures as .
In the second case, the angle is not one of the congruent angles. In this case, the sum of the other two angles is . Therefore the two angles each measure .
Notice that information about the angles of a triangle does not tell us the lengths of the sides. For example, two triangles could have the same three angles, but the triangles are not congruent. That is, the corresponding sides and the corresponding angles do not have the same measures. However, these two triangles will be similar. Next we define similarity and discuss the criteria for triangles to be similar.
Video: Congruent and similar triangles
Consider the situation in which two triangles have three pair of congruent angles.
These triangles are similar. This means that the corresponding angles are congruent, and the corresponding sides are proportional. In the triangles shown above, we have the following:
Example 4: In the triangles shown above, , and . What are the lengths of sides and ?
Solution: and .
Given that , we have .
Similarly, as , we have
Recall that these triangles are considered to be similar because they have three pair of congruent angles. This is just one of three ways to determine that two triangles are similar. The table below summarizes criteria for determining if two triangles are similar.
|AAA||Two triangles are similar of they have three pair of congruent angles|
|SSS||Two triangles are similar if all three pair of corresponding sides are in the same proportion|
|SAS||Two triangles are similar if two pair of corresponding sides are in the same proportion, and the included angles are congruent.|
A special case of SSS is "HL," or "hypotenuse leg." This is the case of two right triangles being similar. This case is examined in example 5 below.
Example 5: Determine if the triangles are similar.
Solution: The triangles are similar
Recall that for every right triangle, we can use the Pythagorean Theorem to find the length of a missing side. In we have:
Similarly, in triangle we have:
Therefore the sides of the triangles are proportional, with a ratio of .
Because we will always be able to use the Pythagorean Theorem in this way, two right triangles will be similar if the hypotenuse and one leg of one triangle are in proportion with the hypotenuse and one leg of the second triangle. This is the HL criteria.
Similar triangles can be used to solve problems in which lengths or distances are proportional. The following example will show you how to solve such problems.
Example 6: Use similar triangles to solve the problem:
A tree casts a shadow that is long. A person who is tall is standing in front of the tree, and his shadow is long. Approximately how tall is the tree?
The picture shows us similar right triangles: the person and his shadow are the legs of one triangle, and the tree and its shadow form the legs of the larger triangle. The triangles are similar because of their angles: they both have a right angle, and they share one angle. Therefore the third angles are also congruent, and the triangles are similar.
The ratio of the triangles' lengths is . If we let represent the height of the tree, we have:
In this lesson we have reviewed key aspects of triangles, including the names of different types of triangles, the triangle angle sum, and criteria for similar triangles. In the last example, we used similar triangles to solve a problem involving an unknown height. In general, triangles are useful for solving such problems, but notice that we did not use the angles of the triangles to solve this problem. This technique will be the focus of problems you will solve later in the chapter.