6.2: Triangles and Angles
Introduction
Tipi Decoration
Jaime has been working hard on her tipi. She has decided to use the following pattern as part of the design on the material. She thinks that if she uses cloth triangles, that she can sew them onto the cloth of the tipi to make a pattern. This red, black, red pattern is going to stretch along the outside bottom edge of the tipi.
“That is very cool,” her sister Lily said admiring Jaime’s sketch.
“I think so too. It should look very pretty on the brown material,” Jaime added.
“Yes. How do you know that the triangles will be the exact same size?”
“That’s easy. I can measure the top angle of the triangle using a protractor and then create each one by tracing. This one will have an angle of \begin{align*}120^{\circ}\end{align*}. when I am done,” Jaime explained to Lily.
“What kind of triangle is that?” Lily asked.
Do you know? This lesson is all about different types of triangles. As you learn all about the different types of triangles, think about this problem. What type of triangle is in the pattern? Can you justify your answer?
What You Will Learn
In this lesson, you will learn how to complete the following skills and tasks.
 Classify triangles by angle measures (acute, right, obtuse) and by side lengths (equilateral, isosceles, scalene)
 Confirm that the sum of any angles of a triangle is equal to \begin{align*}180^{\circ}\end{align*}.
 Describe and analyze triangles and associated angle measures using known information, variable expressions and including triangles formed by two lines intersecting parallel lines.
 Describe and classify triangles found in realworld objects and architecture.
Teaching Time
I. Classify Triangles by Angle Measures (Acute, Right, Obtuse) and by Side Lengths (Equilateral, Isosceles and Scalene)
In this lesson we will examine different kinds of triangles. As we know, triangles are geometric figures that have three sides and three angles. There are different types of triangles too. We can classify or identify them in different ways. One way is by their angles measures and one way is by their sides lengths.
Let’s start with angle measures first.
We all know that triangles have three angles. The corners where each of the line segments connects form an angle. When we know the measures of these angles, we can use this information to name and identify the triangles. Let’s look at some of the different types of triangles according to angle measure.
Acute Triangles are triangles that have three angles that are less than \begin{align*}90^{\circ}\end{align*}. The word “acute” when applied to angles means less than \begin{align*}90^{\circ}\end{align*}, so an acute triangle has three angles where all three angles are less than \begin{align*}90^{\circ}\end{align*}.
Right Triangle a right triangle is a triangle with one \begin{align*}90^{\circ}\end{align*} angle. The other two angles will be acute, but the key to identifying a right triangle is that it has one right angle.
Obtuse Triangle is a triangle with one angle that is obtuse or greater than \begin{align*}90^{\circ}\end{align*}.
Now let’s apply this information in an example.
Example
Identify whether each of the following triangles is acute, obtuse or right.
Now let’s break each one down.
With the first triangle, triangle \begin{align*}a\end{align*}, we can see that one of the angles is greater than 90 degrees. This is an obtuse triangle.
The second triangle has three angles that are less than 90 degrees, this is an acute triangle.
The third triangle also has three angles that are less than 90 degrees. This is also an acute triangle.
The fourth triangle has a right angle. You can see that because it forms a nice neat corner so perfectly. This is a right triangle.
The fifth triangle has an angle greater than 90 degrees. This is an obtuse triangle.
We can also classify or identify triangles by the length of their sides. This means that we look at the line segments that create the triangle.
Equilateral Triangle is a triangle with all three sides equal.
Isosceles Triangle has two sides that are equal in length. Often an isosceles triangle is the trickiest one to identify.
Scalene Triangle is a triangle where none of the sides are the same length. All three sides are different lengths.
Now let’s apply what we have learned and identify some triangles.
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? This 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 last triangle has sides of 5.5 cm, 4.1 cm, and 8 cm. None of the sides are congruent, so this is a scalene triangle.
Sometimes, we can classify a triangle by both its sides and its angles.
Example
Identify each triangle by both its sides and angles.
The first triangle is a right isosceles triangle. It has one right angle and two sides that are the same length.
The second triangle is an acute scalene triangle. All three sides are different lengths and all three angle measures are acute.
The third triangle is an obtuse scalene triangle. It has one obtuse angle and three different side lengths.
The last triangle is an obtuse isosceles. It has one obtuse angle and two side lengths that are the same.
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, the longer the side opposite it is. Therefore we can say that if a triangle has two congruent angles, it must have two congruent sides, and thus it 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.
Notice that the word “congruent” means exactly the same.
II. Confirm that the Sum of any Angles of a Triangle is \begin{align*}\underline{180^{\circ}}\end{align*}
If you look at all of the angles in a triangle, you will notice something consistent about each one of them. If we add up the number of degrees in each angle of a triangle, you will see that the sum of the angle measures is equal to \begin{align*}180^{\circ}\end{align*}.
That is a great question. That short answer is, yes, it is always true. But let’s look at an example to understand this a little further.
We can start by looking at an equilateral triangle. The three angles of an equilateral triangle are all equal. We know from the last section, that each of these angles is 60 degrees. Here is an example of an equilateral triangle.
Now let’s look at what happens when we cut out the \begin{align*}60^{\circ}\end{align*} angles. Three \begin{align*}60^{\circ}\end{align*} angles are equal to \begin{align*}180^{\circ}\end{align*} and there are \begin{align*}180^{\circ}\end{align*} in a straight line. The sum of the angles of a triangle is \begin{align*}180^{\circ}\end{align*}, and this will happen no matter what the angle measures are. The angles of a triangle will always form a straight line and be equal to \begin{align*}180^{\circ}\end{align*}.
Now let’s look at how we can use this information to find missing angle measures.
III. Describe and Analyze Triangles and Associated Angle Measures Using Known Information, Variable Expressions and Triangles Formed by Two Lines Intersecting Parallel Lines
Now that we have looked at how to identify triangles by angle measures and side lengths, we can look at how we can work with missing angle measures. We identify missing angle measures using what we know about triangles and by writing variable expressions. Let’s take a look at how we can problem solve in an example.
Example
What is the missing angle measure of this triangle?
Now we have a triangle here. First, we can use our known information to figure out what kind of triangle it is. Let’s begin by looking at the angles in this triangle. There are two small angles. These are 25 degree angles and they are acute. We can see by looking at this third unknown angle, that is an obtuse angle. This is an obtuse triangle.
Next, we can write an expression to help us to figure out the missing measure.
\begin{align*}25+25+x\end{align*}
Now, we can write this expression into an equation with a sum of 180 degrees. Then we can solve it for the value of \begin{align*}x\end{align*}.
\begin{align*}25+25+x &= 180\\ 50+x &= 180\\ x &= 18050\\ x &= 130^{\circ}\end{align*}
The measure of the missing angle is \begin{align*}130^{\circ}\end{align*}.
Now let’s look at another example.
Example
What is the measure of the two missing angles if this is an isosceles triangle?
Here we have two missing angles. We know from the problem that this is an isosceles triangle. That means that side lengths are the same and we can see that the two base angles are also congruent. Our given angle is \begin{align*}50^{\circ}\end{align*}, so we can write a variable expression to help us figure out the measure of the missing base angles.
\begin{align*}x+x+50\end{align*}
We can expand this expression to an equation that is equal to 180 degrees.
\begin{align*}x+x+50=180\end{align*}
Next, we combine the like terms before solving this.
\begin{align*}2x+50 &= 180\\ 2x &= 130\\ x &= 65^{\circ}\end{align*}
Each of the base angles is equal to \begin{align*}65^{\circ}\end{align*}.
Sometimes, we will have triangles that are created by intersecting lines. When this happens, we can use the information that we know about intersecting lines to figure out the missing angle measures. Let’s look at an example.
Example
Find the value of the missing angles \begin{align*}x\end{align*} and \begin{align*}y\end{align*}
Now to work through this problem, you will need to apply all of the things that you have learned to problem solve the measures of the missing angles.
Let’s start by looking at angle \begin{align*}x\end{align*}.
You can see that angle \begin{align*}x\end{align*} is an acute angle. It is also an adjacent angle with the 140 degree angle already labeled. We know that the sum of adjacent angles is \begin{align*}180^{\circ}\end{align*}. Now we can write an equation and solve for the missing angle measure.
\begin{align*}140 + x &= 180\\ x &= 40^{\circ}\end{align*}
There are two ways to find the measure of angle \begin{align*}y\end{align*}. One is to use the sum of the angle measures given that we know the measure of \begin{align*}x\end{align*}. Let’s do that one first.
\begin{align*}y+40+85 &= 180\\ y+125 &= 180\\ y &= 55^{\circ}\end{align*}
The second way is to use vertical angles. You can see that the angle \begin{align*}125^{\circ}\end{align*} is labeled. This means that the angle vertical to this labeled angle is also \begin{align*}125^{\circ}\end{align*}. The angle \begin{align*}y\end{align*} forms a straight line with that angle and therefore is \begin{align*}55^{\circ}\end{align*}.
See if you can figure out one more way to solve for this missing angle.
IV. Describe and Classify Triangles Found in Real – World Objects and Architecture
There are many triangles out there in the real – world. One of the big places that you will see triangles is in bridge construction. Look at this picture of a truss bridge.
Looking at this bridge, you can see how the basic shape of a triangle is fundamental to the design of the bridge. The triangle helps to keep the bridge stable because of the strength of its foundation. The triangle is a shape that because of its base is very stable and won’t give to pressure. It is a balanced figure.
We can also see triangles in roof design. Take a look at this picture.
The roof construction of this home is using a triangle for stability. It also allows for water runoff during rain or when snow melts. Once again, using triangles makes sense.
Look around your home and neighborhoodwhere are some other places where triangles are very useful? Make a short list of five other places where you can find triangles.
RealLife Example Completed
Tipi Decoration
Here is the original problem once again. Reread it and then answer the questions for the solution.
Jaime has been working hard on her tipi. She has decided to use the following pattern as part of the design on the material. She thinks that if she uses cloth triangles, that she can sew them onto the cloth of the tipi to make a pattern. This red, black, red pattern is going to stretch along the outside bottom edge of the tipi.
“That is very cool,” her sister Lily said admiring Jaime’s sketch.
“I think so too. It should look very pretty on the brown material,” Jaime added.
“Yes. How do you know that the triangles will be the exact same size?”
“That’s easy. I can measure the top angle of the triangle using a protractor and then create each one by tracing. This one will have an angle of \begin{align*}120^{\circ}\end{align*} when I am done,” Jaime explained to Lily.
“What kind of triangle is that?” Lily asked.
Answer the questions on the following page.
Solution to Real – Life Example
Here are the questions from the introduction. What type of triangle is in the pattern? Can you justify your answer?
The triangle in the pattern is an obtuse triangle. It is an obtuse triangle because it has an angle that is greater than \begin{align*}90^{\circ}\end{align*}. The other two angles are less than \begin{align*}90^{\circ}\end{align*}. Because the largest angle is \begin{align*}120^{\circ}\end{align*}, it is an obtuse triangle.
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Acute Triangle
 A triangle where all three angles are less than \begin{align*}90^{\circ}\end{align*}.
 Right Triangle
 A triangle with one \begin{align*}90^{\circ}\end{align*} angle and two acute angles.
 Obtuse Triangle
 a triangle with one angle that is greater than \begin{align*}90^{\circ}\end{align*}.
 Equilateral Triangle
 all three side lengths and all three angles are congruent.
 Isosceles Triangle
 two side lengths are the same.
 Scalene Triangle
 all three side lengths are different
 Congruent
 means exactly the same, having the same measure.
Time to Practice
Directions: Classify each triangle by the describe angle measures as acute, obtuse or right.
 A triangle with three \begin{align*}60^{\circ}\end{align*} angles.
 A triangle with one \begin{align*}110^{\circ}\end{align*} angle.
 A triangle with one right angle and two acute angles.
 A triangle with one \begin{align*}130^{\circ}\end{align*} angle.
 A triangle with three acute angles.
Directions: Identify each triangle by the side lengths described. Identify them as equilateral, isosceles or scalene.
 A triangle with side lengths of 6 in, 6 in and 4 inches.
 A triangle with side lengths of 3 ft, 4 ft, and 5 ft.
 A triangle with side lengths of 8 inches.
 A triangle with side lengths of 7 inches, 8 inches and 8 inches.
 A triangle with side lengths of 6 meters, 8 meters and 10 meters.
 A triangle with side lengths of 10 mm.
Directions: Using what you have learned about the interior angles of a triangle, determine the missing angle in each triangle.
 \begin{align*}45^{\circ}, 45^{\circ}, ?\end{align*}
 \begin{align*}60^{\circ}, 60^{\circ}, ?\end{align*}
 \begin{align*}90^{\circ}, 50^{\circ}, ?\end{align*}
 \begin{align*}100^{\circ}, 40^{\circ}, ?\end{align*}
 \begin{align*}110^{\circ}, 30^{\circ}, ?\end{align*}
 \begin{align*}50^{\circ}, 10^{\circ}, ?\end{align*}
Directions: Identify three triangles in the room around you.



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