8.3: Triangles
Introduction
The Triangle Sculpture
Kevin and Jake began examining a sculpture while the girls were examining the painting with the lines. This sculpture is full of triangles. The boys remember how Mrs. Gilson explained that a triangle is one of the strongest figures that there is and that is why we see triangles in construction.
“Think of a bridge,” Kevin said to Jake. “A bridge has many triangles within it. That is how the whole thing stays together. If it did not have the triangles, the structure could collapse.”
“What about here? Do you think it matters what kind of triangle is used?” Jake asked.
“I don’t know. Let’s look at what they used here.”
The two boys walked around the sculpture and looked at it from all sides. There was a lot to notice. After a little while, Jake was the first one to speak.
“I don’t think it matters which triangle you use,” he said.
“Oh, I do. The isosceles makes the most sense because it is balanced,” Kevin said smiling.
Jake is confused. He can’t remember why an isosceles triangle “is balanced” in Kevin’s words. Jake stops to think about this as Kevin looks at the next sculpture.
Do you know what Kevin means? What is an isosceles triangle and how does it “balance?” This lesson will teach you all about triangles and how to classify them. When you finish with this lesson, you will have a chance to revisit this problem. Then you may understand a little better what Kevin means by his words.
What You Will Learn
In this lesson you will learn how to demonstrate the following skills:
- Recognize the sum of the interior angles of a triangle as \begin{align*}180^\circ\end{align*}, and the supplementary relationship between an interior angle and its adjacent exterior angle.
- Classify triangles by angle measures.
- Classify triangles by side lengths.
- Describe and analyze triangles and associated angle measures using known classifications and sufficient given information.
Teaching Time
I. Recognize the Sum of the Interior Angles of a Triangle as \begin{align*}180^\circ\end{align*}, and the Supplementary Relationship between an Interior Angle and its Adjacent Exterior Angle
This lesson is all about triangles. You have been learning about triangles for a long time. It is one of the first shapes that small children learn to recognize. Mathematically speaking, we know that the prefix “tri” means three and the rest of the word is “angles.” Therefore, a triangle is a figure with three sides and three angles.
In a triangle there is a relationship between the interior angles of the triangle. What are interior angles?
Interior angles are the angles inside the triangle. There are three of them and we can learn about the relationship between the interior angles of a triangle by looking at a few examples.
Notice that triangles \begin{align*}a, b, c\end{align*} and \begin{align*}d\end{align*} are all different, they have different angle measures and different side lengths. Look closely, though. If you add up the measures of the three angles, they always equal \begin{align*}180^\circ\end{align*}!
Write this down in your notebooks.
Now let’s look at triangles a little differently. In geometry, a triangle can be formed by the intersection of three lines.
First, notice that the lines create the three interior angles of the triangle. And as we know, those three angles have a sum of \begin{align*}180^\circ\end{align*}.
Next notice that if we extend any side of the triangle, then it stretches beyond the triangle. Now we have a pair of angles, an interior angle and an exterior angle.
An exterior angle is the angle formed outside of the edge of the triangle.
Here is a clearer example of an exterior angle.
As you can see, the interior angle and the exterior angle form a line. Therefore their sum must be \begin{align*}180^\circ\end{align*}.
The adjacent angle to the interior angle is \begin{align*}120^\circ\end{align*}. If the exterior and the interior angle form a straight line, then their sum is \begin{align*}180^\circ\end{align*}. We can set up an equation and solve for the measure of \begin{align*}x\end{align*}.
\begin{align*}120 + x &= 180\\ x &= 180 - 120\\ x &= 60^\circ\end{align*}
The missing measure of the interior angle is \begin{align*}60^\circ\end{align*}.
We could also figure this out another way. Take a look at the other given interior angles of the triangle. They are \begin{align*}50^\circ\end{align*} and \begin{align*}70^\circ\end{align*}. Their sum is also \begin{align*}120^\circ\end{align*}!
In fact, the sum of any two interior angles in a triangle is always equal to the exterior angle of the third angle.
But, we can use this information to figure out the missing third interior angle. If the sum of the two interior angles is 120, we can use the same equation to solve for the third missing angle.
\begin{align*}120 + x &= 180\\ x &= 180 -120\\ x &= 60^\circ\end{align*}
Notice that both methods will help you to find the correct measure of a missing interior angle.
Now let’s apply this to an example.
Example
What is the measure of angle \begin{align*}S\end{align*} in the figure below?
We can see that angle \begin{align*}S\end{align*} is an exterior angle. However, we do not know what its adjacent interior angle is. Can we still find the measure of angle \begin{align*}S\end{align*}? We can. As we have just learned, the other two interior angles have a sum equal to the measure of the third angle’s exterior angle. Therefore angle \begin{align*}S\end{align*} must be equal to the sum of the two angles we have been given.
\begin{align*}S &= 30 + 35\\ S &= 65^\circ\end{align*}
Incidentally, we can also find the measure of the third angle in the triangle by using the exterior angle. We know that the sum of this angle and angle \begin{align*}S\end{align*} must be \begin{align*}180^\circ\end{align*}. If \begin{align*}S\end{align*} is \begin{align*}65^\circ\end{align*}, then the angle must be \begin{align*}180 - 65 = 115^\circ\end{align*}.
We also know that the sum of the three interior angles of a triangle is \begin{align*}180^\circ\end{align*}, so we could also find the missing angle by adding the two known angles and then subtracting from \begin{align*}180^\circ\end{align*}.
\begin{align*}\angle 1 + \angle 2 + \angle 3 &= 180^\circ\\ 30 + 35 + \angle 3 &= 180^\circ\\ 65 + \angle 3 &= 180^\circ\\ \angle 3 &= 180 - 65\\ \angle 3 &= 115^\circ\end{align*}
8E. Lesson Exercises
Use what you have learned to answer each question.
- If the sum of two angles of a triangle is \begin{align*}150^\circ\end{align*}, then what is the value of the third angle?
- If the sum of two of the angles is \begin{align*}75^\circ\end{align*}, then what is the measure of the third angle’s exterior angle?
- Angle \begin{align*}A = 33^\circ\end{align*}, Angle \begin{align*}B = 65^\circ\end{align*}, what is the measure of Angle \begin{align*}C\end{align*}?
Take a few minutes to check your work with a friend.
II. Classify Triangles by Angle Measures
Now that you understand the angles of triangles, let’s look at classifying them according to angles.
As we have seen, the angles in a triangle can vary a lot in size and shape, but they always total \begin{align*}180^\circ\end{align*}. We can identify kinds of triangles by the sizes of their angles. A triangle can either be acute, obtuse, or right. Let’s look more closely.
Acute triangles have three acute angles. In other words, all of their angles measure less than \begin{align*}90^\circ\end{align*}. Below are some examples of acute triangles.
Notice that each angle in the triangles above is less than \begin{align*}90^\circ\end{align*}, but the total for each triangle is still \begin{align*}180^\circ\end{align*}.
We classify triangles that have an obtuse angle as an obtuse triangle. This means that one angle in the triangle measures more than \begin{align*}90^\circ\end{align*}. Here are some obtuse triangles.
You can see that obtuse triangles have one wide angle that is greater than \begin{align*}90^\circ\end{align*}. Still, the three angles in obtuse triangles always add up to \begin{align*}180^\circ\end{align*}. Only one angle must be obtuse to make it an obtuse triangle.
The third kind of triangle is a right triangle. As their name implies, right triangles have one right angle that measures exactly \begin{align*}90^\circ\end{align*}. Often, a small box in the corner tells you when an angle is a right angle. Let’s examine a few right triangles.
Once again, even with a right angle, the three angles still total \begin{align*}180^\circ\end{align*}!
Now let’s practice identifying each kind of triangle.
Example
Label each triangle as acute, obtuse, or right.
In order to classify the triangles, we must examine the three angles in each. If there is an obtuse angle, it is an obtuse triangle. If there is a right angle, it is a right triangle. If all three angles are less than \begin{align*}90^\circ\end{align*}, it is an acute triangle.
One short cut we can use is to compare the angles to \begin{align*}90^\circ\end{align*}. If an angle is exactly \begin{align*}90^\circ\end{align*}, we know the triangle must be a right triangle. If any angle is more than \begin{align*}90^\circ\end{align*}, the triangle must be an obtuse triangle. If there are no right or obtuse angles, the triangle must be an acute triangle. Check to make sure each angle is less than \begin{align*}90^\circ\end{align*}.
There are no right angles in Figure 1. There are no angles measuring more than \begin{align*}90^\circ\end{align*}. This is probably an acute triangle. Let’s check each angle to be sure: \begin{align*}30^\circ, 70^\circ,\end{align*} and \begin{align*}80^\circ\end{align*} are all less than \begin{align*}90^\circ,\end{align*} so this is definitely an acute triangle. Figure 1 is an acute triangle.
Is there any right or obtuse angles in the second triangle? The small box in the corner tells us that the angle is a right angle. Therefore this is a right triangle. Figure 2 is a right triangle.
What about Figure 3? It does not have any right angles. It does, however, have an extremely wide angle. Wide angles usually are obtuse. Let’s check the measure to make sure it is more than \begin{align*}90^\circ\end{align*}. It is \begin{align*}140^\circ\end{align*}, so it is definitely an obtuse angle. Therefore this is an obtuse triangle. Figure 3 is an obtuse triangle.
Figure 4 doesn’t have any right angles. It doesn’t have any wide angles either. Obtuse angles are not always wide, however. Check the angle measures to be sure. The angle measuring \begin{align*}95^\circ\end{align*} is greater than \begin{align*}90^\circ,\end{align*} so it is obtuse. That makes this an obtuse triangle. Figure 4 is an obtuse triangle.
Yes. Make a few notes about each type of triangle so that you can remember how to classify them according to their angles.
8F. Lesson Exercises
Determine the type of triangles described in each example.
- One angle is \begin{align*}103^\circ\end{align*} and the other two angles are acute angles.
- Two out of three angles measure \begin{align*}45^\circ\end{align*}.
Those were a little like a riddle. Check your answers with a friend and then continue.
III. Classify Triangles by Side Lengths
You have seen that we can classify triangles by their angles. We can also classify triangles by the lengths of their sides. As you know, every triangle has three sides. Sometimes all three sides are the same length, or congruent. In some triangles, only two sides are congruent. And still other triangles have sides that are all different lengths. By comparing the lengths of the sides, we can determine the kind of triangle it is.
Let’s see how this works.
A triangle with three equal sides is an equilateral triangle. It doesn’t matter how long the sides are, as long as they are all congruent, or equal. Here are some equilateral triangles.
Just remember, equal sides means it’s an equilateral triangle.
An isosceles triangle has two congruent sides. It doesn’t matter which two sides, any two will do. Let’s look at some isosceles triangles.
The third type of triangle is a scalene triangle. In a scalene triangle, none of the sides are congruent.
Now let’s practice identifying each kind of triangle.
Example
Classify each triangle as equilateral, isosceles, or scalene.
We need to examine the lengths of the sides in each triangle to see if any sides are congruent. In the first triangle, two sides are 7 meters long, but the third side is shorter. Which kind of triangle has two congruent sides? The first triangle is an isosceles triangle.
Now let’s look at the second triangle. All three sides are the same length, so this must be an equilateral triangle. The second triangle is an equilateral triangle.
The last triangle has sides of 5 cm, 4 cm, and 8 cm. None of the sides are congruent, so this is a scalene triangle. The last triangle is a scalene triangle.
8G. Lesson Exercises
Classify each triangle by its description.
- Two side lengths are 6 in, one side length is 8 in.
- All three side lengths are 10 mm.
- All three side lengths have different measures.
Take a few minutes to go over your responses with a partner.
We have seen that we can classify triangles either by their angles or by their sides. Now let’s make things a bit trickier...we can do both at the same time! For example, an isosceles triangle that has one right angle is a right isosceles triangle. A scalene triangle with angles that are all less than \begin{align*}90^\circ\end{align*} is an acute scalene triangle. Let’s look below at some other examples.
As you can see, the first name identifies the triangle by its angles, and the second name groups it by its sides. Equilateral triangles do not quite fit this pattern. They are always acute. This is because the three angles in an equilateral triangle always measure \begin{align*}60^\circ\end{align*}.
There is one more thing to know about classifying triangles by their angles and sides. We can also tell whether a triangle is isosceles, scalene, or equilateral by its angles. Every angle is related to the side opposite it. Imagine a book opening. The wider you open it, the greater the distance between the two flaps. In other words the wider an angle is, the longer the opposite side.
Therefore we can say that if a triangle has two congruent angles, it must have two congruent sides, and must be isosceles. If it has three angles of different measures, then its sides are also all of different lengths, so it is scalene. Finally, an equilateral triangle, as we have seen, always has angles of \begin{align*}60^\circ,\end{align*} and these angles are opposite congruent sides.
IV. Describe and Analyze Triangles and Associated Angle Measures using Known Classifications and Sufficient Given Information
When lines intersect in the geometric plane, they often form triangles. We can apply what we know to these triangles in order to classify them, find side lengths, or solve for unknown angle measures. Take a look at the diagram below.
As you can see, three lines have intersected to form triangle \begin{align*}ABC\end{align*}.
Can we classify this triangle? We can, but we’ll need to do some analysis first. Let’s use what we know about triangles to interpret the information we have been given. So far, we know the measure of only one angle in the triangle, angle \begin{align*}A\end{align*}. The only other angle we know is adjacent to angle \begin{align*}B\end{align*}. How can we use this information?
We need to keep in mind one of the essential rules of plane geometry: a line is really a straight angle, with a measure of \begin{align*}180^\circ\end{align*}. That means that the \begin{align*}120^\circ\end{align*} angle and angle \begin{align*}B\end{align*} are supplements. In other words, they must add up to \begin{align*}180^\circ\end{align*} because together they form a line. So now we can set up an equation to solve for angle \begin{align*}B\end{align*}.
\begin{align*}\angle B + 120 &= 180^\circ\\ \angle B &= 180 - 120\\ \angle B &= 60^\circ\end{align*}
Fill this information into the diagram. Now we know two of the angles in triangle \begin{align*}ABC\end{align*}. That means we can solve for the third. Remember, the three angles in a triangle always have a sum of \begin{align*}180^\circ\end{align*}. Let’s set up another equation to solve for angle \begin{align*}C\end{align*}.
\begin{align*}\angle A + \angle B + \angle C &= 180^\circ\\ 60 + 60 + \angle C &= 180^\circ\\ 120 + \angle C &= 180^\circ\\ \angle C &= 60^\circ\end{align*}
Now we know that each angle in triangle \begin{align*}ABC\end{align*} measures \begin{align*}60^\circ\end{align*}. Now we can identify this triangle. What kind is it? Any triangle with three equal angles (which are always \begin{align*}60^\circ\end{align*}) is an equilateral triangle.
Figuring out triangles like this one is a lot like being a detective! You have to use the clues or given information to solve the riddle!
Real Life Example Completed
The Triangle Sculpture
Here is the original problem once again. Reread it and underline any important information.
Kevin and Jake began examining a sculpture while the girls were examining the painting with the lines. This sculpture is full of triangles. The boys remember how Mrs. Gilson explained that a triangle is one of the strongest figures that there is and that is why we see triangles in construction.
“Think of a bridge,” Kevin said to Jake. “A bridge has many triangles within it. That is how the whole thing stays together. If it did not have the triangles, the structure could collapse.”
“What about here? Do you think it matters what kind of triangle is used?” Jake asked.
“I don’t know. Let’s look at what they used here.”
The two boys walked around the sculpture and looked at it from all sides. There was a lot to notice. After a little while, Jake was the first one to speak.
“I don’t think it matters which triangle you use,” he said.
“Oh, I do. The isosceles makes the most sense because it is balanced,” Kevin said smiling.
Jake is confused. He can’t remember why an isosceles triangle “is balanced” in Kevin’s words. Jake stops to think about this as Kevin looks at the next sculpture.
Kevin’s comment is a little tricky. You can think of an isosceles triangle as being balanced because it has two equal sides. Therefore, if you look at an isosceles triangle, it will be even whereas a scalene triangle would not be. In Kevin’s thinking, this type of triangle makes sense because it would be firm, solid and “balanced.”
If you think about Kevin’s statement, you can grasp the math by thinking about the properties of an isosceles triangle. Look at the sculpture again. How are triangles being used in the sculpture? Can you see any other types of triangles in this sculpture? Make a few notes in your notebook.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Triangle
- a figure with three sides and three angles.
- Interior angles
- the three inside angles of a triangle.
- Exterior angles
- the angles outside of a triangle formed by intersecting lines
- Acute Triangles
- triangles with three angles less than \begin{align*}90^\circ\end{align*}
- Obtuse Triangles
- triangle with one angle greater than \begin{align*}90^\circ\end{align*}
- Right Triangle
- a triangle with one \begin{align*}90^\circ\end{align*} angle.
- Congruent
- exactly the same
- Equilateral Triangle
- all three side lengths are the same
- Isosceles Triangle
- two side lengths are the same and one is different.
- Scalene Triangle
- all three side lengths of a triangle are different lengths.
Technology Integration
Khan Academy Acute, Right, and Obtuse Angles
James Sousa, Angle Relationships and Types of Triangles
Other Videos:
- http://www.mathplayground.com/mv_triangle_angle_sum.html – This is a Brightstorm video that demonstrates how the sum of the interior angles is always 180 degrees.
Time to Practice
Directions: Find the measure of angle \begin{align*}H\end{align*} in each figure below.
Directions: Identify each triangle as right, acute, or obtuse.
Directions: Identify each triangle as equilateral, isosceles, or scalene.
Directions: Use what you have learned to answer each question.
13. True or false. An acute triangle has three sides that are all different lengths.
14. True or false. A scalene triangle can be an acute triangle as well.
15. True or false. An isosceles triangle can also be a right triangle.
16. True or false. An equilateral triangle has three equal sides.
17. True or false. An obtuse triangle can have multiple obtuse angles.
18. True or false. A scalene triangle has three angles less than 90 degrees.
19. True or false. A triangle with a \begin{align*}100^\circ\end{align*} angle must be an obtuse triangle.
20. True or false. The angles of an equilateral triangle are also equal in measure.