Triangles

 

Learning Objective(s)

·         Identify equilateral, isosceles, scalene, acute, right, and obtuse triangles.

·         Identify whether triangles are similar, congruent, or neither.

·         Identify corresponding sides of congruent and similar triangles.

·         Find the missing measurements in a pair of similar triangles.

·         Solve application problems involving similar triangles.

Geometric shapes, also called figures, are an important part of the study of geometry. The triangle is one of the basic shapes in geometry. It is the simplest shape within a classification of shapes called polygons. All triangles have three sides and three angles, but they come in many different shapes and sizes. Within the group of all triangles, the characteristics of a triangle’s sides and angles are used to classify it even further. Triangles have some important characteristics, and understanding these characteristics allows you to apply the ideas in real-world problems.

 

A polygon is a closed plane figure with three or more straight sides. Polygons each have a special name based on the number of sides they have. For example, the polygon with three sides is called a triangle because “tri” is a prefix that means “three.” Its name also indicates that this polygon has three angles. The prefix “poly” means many.

 

The table below shows and describes three classifications of triangles. Notice how the types of angles in the triangle are used to classify the triangle.

 

 

The sum of the measures of the three interior angles of a triangle is always 180°. This fact can be applied to find the measure of the third angle of a triangle, if you are given the other two. Consider the examples below.

 

 

 

 

 

 

There is an established convention for naming triangles. The labels of the vertices of the triangle, which are generally capital letters, are used to name a triangle.

 

 

You can call this triangle ABC or  since A, B, and C are vertices of the triangle. When naming the triangle, you can begin with any vertex. Then keep the letters in order as you go around the polygon. The triangle above could be named in a variety of ways: , or . The sides of the triangle are line segments AB, AC, and CB.

 

Just as triangles can be classified as acute, obtuse, or right based on their angles, they can also be classified by the length of their sides. Sides of equal length are called congruent sides. While we designate a segment joining points A and B by the notation , we designate the length of a segment joining points A and B by the notation AB without a segment bar over it. The length AB is a number, and the segment  is the collection of points that make up the segment.

 

Mathematicians show congruency by putting a hash mark symbol through the middle of sides of equal length. If the hash mark is the same on one or more sides, then those sides are congruent. If the sides have different hash marks, they are not congruent. The table below shows the classification of triangles by their side lengths.

 

 

To describe a triangle even more specifically, you can use information about both its sides and its angles. Consider this example.

 

 

 

 

Classify the triangle shown below.     A) acute scalene B) right isosceles C) obtuse scalene D) obtuse isosceles   Show/Hide Answer A) acute scalene Incorrect. This triangle has one angle (angle Q) that is between 90º and 180º, so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.   B) right isosceles Incorrect. This triangle does not contain a right angle. It has one angle (angle Q) that is between 90º and 180º, so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.   C) obtuse scalene Correct. This triangle has vertices P, Q, and R, one angle (angle Q) that is between 90º and 180º, and sides of three different lengths.   D) obtuse isosceles Incorrect. Although this triangle is obtuse, it does not have two sides of equal length. Its three sides are all different lengths, so it is scalene. The correct answer is obtuse scalene.

 

 

Two triangles are congruent if they are exactly the same size and shape. In congruent triangles, the measures of corresponding angles and the lengths of corresponding sides are equal. Consider the two triangles shown below:

 

 

Since both  and  are right angles, these triangles are right triangles. Let’s call these two triangles  and . These triangles are congruent if every pair of corresponding sides has equal lengths and every pair of corresponding angles has the

same measure.

 

The corresponding sides are opposite the corresponding angles.

 

 

 

 and are congruent triangles as the corresponding sides and corresponding angles are equal.

 

Let’s take a look at another pair of triangles. Below are the triangles  and .

 

 

These two triangles are surely not congruent because is clearly smaller in size than . But, even though they are not the same size, they do resemble one another. They are the same shape. The corresponding angles of these triangles look like they might have the same exact measurement, and if they did they would be congruent angles and we would call the triangles similar triangles.

 

Congruent angles are marked with hash marks, just as congruent sides are.

 

Image showing angle measurements of both triangles.	Image showing triangles ABC and RST using hash marks to show angle congruency.

 

We can also show congruent angles by using multiple bands within the angle, rather than multiple hash marks on one band.  Below is an image using multiple bands within the angle.

 

Image showing triangles ABC and RST

using bands to show angle congruency.

 

If the corresponding angles of two triangles have the same measurements they are called similar triangles. This name makes sense because they have the same shape, but not necessarily the same size. When a pair of triangles is similar, the corresponding sides are proportional to one another. That means that there is a consistent scale factor that can be used to compare the corresponding sides. In the previous example, the side lengths of the larger triangle are all 1.4 times the length of the smaller. So, similar triangles are proportional to one another.

 

Just because two triangles look similar does not mean they are similar triangles in the mathematical sense of the word. Checking that the corresponding angles have equal measure is one way of being sure the triangles are similar.

 

 

There is another method for determining similarity of triangles that involves comparing the ratios of the lengths of the corresponding sides.

 

If the ratios of the pairs of corresponding sides are equal, the triangles are similar.

 

 

Consider the two triangles below.

 

 

 is not congruent to  because the side lengths of  are longer than those of . So, are these triangles similar? If they are, the corresponding sides should be proportional.

 

Since these triangles are oriented in the same way, you can pair the left, right, and bottom sides:  and ,  and ,  and . (You might call these the two shortest sides, the two longest sides, and the two leftover sides and arrived at the same ratios). Now we will look at the ratios of their lengths.

 

 

Substituting the side length values into the proportion, you see that it is true:

 

 

If the corresponding sides are proportional, then the triangles are similar. Triangles ABC and DEF are similar, but not congruent.

 

Let’s use this idea of proportional corresponding sides to determine whether two more triangles are similar.

 

 

 

 

The mathematical symbol ~ means “is similar to”. So, you can write  is similar to  as .

 

 

Determine whether the two triangles are similar, congruent, or neither.     A)  and  are congruent. B)  and  are similar. C)  and  are similar and congruent. D)  and  are neither similar nor congruent.   Show/Hide Answer A)  and  are congruent. Incorrect. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. They are the same exact size and shape.  is equilateral and  is isosceles, so they are not the same exact shape. The correct answer is  and are neither similar nor congruent.   B) and  are similar. Incorrect. The ratios of the corresponding sides are not equal, so the triangles cannot be similar: . The correct answer is  and are neither similar nor congruent.   C)  and  are similar and congruent. Incorrect. All congruent triangles are similar, but these triangles are not congruent. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure.  is equilateral and  is isosceles, so they are not the same exact shape. The correct answer is  and are neither similar nor congruent.   D)  and  are neither similar nor congruent. Correct. The corresponding angle measures are not known to be equal as shown by the absence of congruence marks on the angles. Also, the ratios of the corresponding sides are not equal: .

 

 

 

You can find the missing measurements in a triangle if you know some measurements of a similar triangle. Let’s look at an example.

 

 

 

This process is fairly straightforward—but be careful that your ratios represent corresponding sides, recalling that corresponding sides are opposite corresponding angles.

 

Applying knowledge of triangles, similarity, and congruence can be very useful for solving problems in real life. Just as you can solve for missing lengths of a triangle drawn on a page, you can use triangles to find unknown distances between locations or objects.

 

Let’s consider the example of two trees and their shadows. Suppose the sun is shining down on two trees, one that is 6 feet tall and the other whose height is unknown. By measuring the length of each shadow on the ground, you can use triangle similarity to find the unknown height of the second tree.

 

First, let’s figure out where the triangles are in this situation! The trees themselves create one pair of corresponding sides. The shadows cast on the ground are another pair of corresponding sides. The third side of these imaginary similar triangles runs from the top of each tree to the tip of its shadow on the ground. This is the hypotenuse of the triangle.

 

If you know that the trees and their shadows form similar triangles, you can set up a proportion to find the height of the tree.

 

 

 

 

 

Triangles are one of the basic shapes in the real world. Triangles can be classified by the characteristics of their angles and sides, and triangles can be compared based on these characteristics. The sum of the measures of the interior angles of any triangle is 180º. Congruent triangles are triangles of the same size and shape. They have corresponding sides of equal length and corresponding angles of the same measurement. Similar triangles have the same shape, but not necessarily the same size. The lengths of their sides are proportional. Knowledge of triangles can be a helpful in solving real-world problems.