Alternative Format — Lesson 2: Angle Properties of Triangles

Let's Start Thinking

Triangles

Triangles are important shapes. Triangles are used often in the following scenarios:

• Construction or engineering because of the stability that they provide. 
  

  Source: Roof Structure - SUNG YOON JO/iStock/Getty Images

• Trigonometry because properties of right triangles are used to calculate unknown distances.
  

  Source: Train - ioanmasay/iStock/Getty Images

Understanding the angle relationships in triangles will prepare you for the study of trigonometry later in your math lessons. Understanding the angle relationships in triangles also helps us understand angle relationships in other polygons. This is because all other polygons can be constructed by joining triangles.

In this lesson, we are going to focus on the properties of the interior and exterior angles of a triangle. We will solve problems using these properties.

Try This

In an isosceles triangle, one of the angle measures is twice the other. What type of triangle could this be (acute, right, or obtuse)? 

Classifying Triangles

Triangles

A triangle is made of three points, called vertices, connected by three line segments, called edges (or sides). 

The name of a triangle consists of its three vertices.

For example, this triangle is called \(\triangle ABC\) (although it could also be called \(\triangle BAC\), or any other possible ordering of the vertices). 

The three angles which lie on the inside of the triangle, at each of the vertices, are called interior angles. 

For example, the acute angles \(\angle ABC\), \(\angle BAC\) and \(\angle ACB\) are the interior angles of this triangle. 

Classifying Triangles by Side Lengths 

One way to classify triangles is by examining their side lengths. There are three types of triangles that arise when we do this:

• An equilateral triangle, which has all three sides equal in length,

  

• An isosceles triangle, which has two sides equal in length, and

  

• A scalene triangle, which has three sides that are all different in length.

  

Since the definition of an isosceles triangle does not say that exactly two sides are equal in length, all equilateral triangles can be considered as isosceles as well. That is, an equilateral triangle is a special case of an isosceles triangle. 

Note: On a diagram, any sides on a triangle that have the same number of tick marks are equal in length.

For example, \(\triangle ABC\) is equilateral, since all three side lengths are equal.

\(\triangle DEF\) is isosceles, since two side lengths are equal. 

A triangle with no tick marks on any of the sides means that none of the side lengths are known to be equal.

Classifying Triangles by Angles 

Another way to classify triangles is by examining the measure of the interior angles. Again, there are three types of triangle:

• An acute triangle, which has interior angles that are all acute (i.e., all less than \(90^\circ\)).

  

• A right triangle, which has one interior angle of \(90^\circ\) (i.e., one right angle).

  

• An obtuse triangle, which has one obtuse interior angle (i.e., one interior angle greater than \(90^\circ\)).

  

These classifications will be used frequently throughout this lesson.

The Interior Angles of a Triangle

In this section, we will be exploring the relationship between the three interior angles of a triangle.

Explore This 1

Description

In the following investigation, what is the relationship between angles in a triangle? Do you notice something regardless of the size, and/or shape of the triangle that you create? Also notice what happens to the sum of the interior angles.

The following are examples of different triangles you could have created. 

Example 1

Sum of interior angles: \(78^\circ + 66^\circ + 36^\circ= 180^\circ\)

Example 2

Sum of interior angles: \(64^\circ + 90^\circ + 26^\circ= 180^\circ\)

Example 3

Sum of interior angles: \(14^\circ + 16^\circ + 150^\circ= 180^\circ\)

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Interactive Version

Interior Angles in a Triangle

Explore This Summary

You may have noticed that, no matter what type of triangle you constructed, the three interior angles of the triangle always added to \(180^\circ\).

Sum of the Interior Angles of a Triangle

In the first Explore This activity, we saw an important fact about triangles.

You can verify this for yourself by creating a triangle on paper and cutting it out.

Mark each of the interior angles so you can keep track of them

If you rip or cut the corners off of the triangle,

you can reassemble them so that the angles all share the same vertex.

You will find that, all together, the three angles form a straight line, and so they must add to \(180^\circ\).

Justifying the Sum of Interior Angles

The fact that the sum of the interior angles of a triangle is \(180^\circ\) can be justified using some other geometric relationships.

Start by drawing \(\triangle ABC\). We can label the measures of interior angles of this triangle using \(a^\circ\) at vertex \(A\), \(b^\circ\) at vertex \(B\), and \(c^\circ\) at vertex \(C\).

Draw a line segment \(PQ\) through vertex \(A\) that is parallel to edge of the triangle \(BC\).

Now the figure contains two parallel line segments, with two of the sides of the triangle, \(AB\) and \(AC\), acting as transversals. That means that we can use the properties of parallel lines in a transversal to help answer this question. 

Side \(AB\), along with the parallel line segments, creates a pair of alternate angles, interior \(\angle b\) and \(\angle PAB\).

Alternate angles are equal, so we can mark the measure of \(\angle PAB\) as \(b^\circ\), as well.

Side \(AC\) also creates a pair of alternate angles, interior \(\angle c\) and \(\angle QAC\). 

These two angles are equal, so we can mark the measure of \(\angle QAC\) as \(c^\circ\) too.

Looking at line segment \(PQ\), we can see that the angles marked \(b^\circ\), \(a^\circ\), and \(c^\circ\) share a vertex and form a straight line. 

That means that the measures of these angles must add to \(180^\circ\). So 

\(a+b+c=180\)

But these angles are also the angles that were originally inside the triangle.

So we have just shown that the sum of the interior angles of a triangle is \(180^\circ\).

Now let's use this property of the interior angles of a triangle to determine the measures of some unknown angles.

Example 1

Solution

Although it's possible to answer this question without drawing a diagram, that's always a good strategy to use for a geometry problem. So let's draw a triangle with one angle measuring \(79^\circ\) and a second angle measuring \(47^\circ\).

We can use a variable, such as \(d^\circ\), to represent the measure of the unknown angle.

Given that the sum of the interior angles of a triangle must be \(180^\circ\), we can say that

\( 79+47+d=180 \)

Simplifying and solving, we find that

\(\begin{align*} 79+47+d&=180 \\ 126 + d &\;= 180 \\ d &\;= 54 \end{align*}\)

Therefore, the third interior angle measures \(54^\circ\).

Alternatively:

We can also find the measure of the third interior angle without an equation by subtracting the two known angles from \(180^\circ\). We get that

\(180^\circ-79^\circ-47^\circ=54^\circ\)

Can a Triangle Have More than One Right Angle?

You may have wondered why just one right angle. Can a triangle have more than one right angle? Let's use what we know about sum of the interior angles of a triangle to answer this question. Let’s investigate.

Consider \(\triangle PQR\).

Let's assume that there are, in fact, two right angles in this triangle. So \(\angle P=90^\circ\) and \(\angle Q = 90^\circ\).

Recall that

\(\angle P + \angle Q + \angle R = 180^\circ \)

Since \(\angle P\) and \(\angle Q\) are both \(90^\circ\),  \(\angle P+\angle Q = 180^\circ\). 

This means that \(\angle R = 0^\circ\).

It's not possible for a triangle to have an angle of \(0^\circ\) at any of its vertices. If it did, it would just be two sides, one on top of the other. Since this condition is not possible, our assumption that there could be twp \(90^\circ\) angles must have been wrong.

Therefore, a triangle cannot have more than one right angle. 

Note that a similar argument can be made for obtuse triangles, which contains one obtuse angle. A triangle cannot have more than one obtuse angle, either.

Check Your Understanding 1

Question

\(\triangle ABC\) has two known interior angles, \(\angle ABC = 24^\circ\) and \(\angle BAC = 125^\circ\). What is the measure of the missing interior angle, \(\angle ACB\)? 

Answer

The measure of \(\angle ACB\) is \(31^\circ\).

Feedback

Let \(d^\circ\) represent the missing angle, \(\angle ACB\).

The sum of the interior angles of a triangle is \(180^\circ\), so we can solve the following equation:

\(\begin{align*} d+125+24 &= 180 \\ d + 149 &= 180\\ d &= 31\end{align*}\)

Therefore, the measure of \(\angle ACB\) is \(31^\circ\).

Example 2

Solution

Since the line through \(B\) and \(C\) is parallel to the line through \(D\) and \(E\), \(\angle ABC\) is corresponding to \(\angle ADE\) (and the two angles are equal). 

Therefore,  \(\angle ADE = 75^\circ\).

Now consider the larger triangle, \(\triangle ADE\).  \(\angle BAC\) is one of the interior angles of this triangle and we know the measures of the other two interior angles:  \(\angle D=75^\circ\) and \(\angle E=80^\circ\).

Since the interior angles of a triangle add to \(\)\(180^\circ\), \(\angle BAC=180^\circ - 75^\circ-80^\circ\). 

Thus, \(\angle BAC=25^\circ\). 

Note that it would have also been possible to solve this question by finding \(\angle ACB\) first, then considering the smaller triangle \(\triangle ABC\). 

Example 3

Solution

To classify the triangle as acute, right, or obtuse, we need to know the measures of the interior angles.

To determine the measures of the interior angles, we need to determine the value of \(z\).

We know that the interior angles must have \(\angle R + \angle S + \angle T= 180^\circ\), and so \(z + \left(\dfrac12z\right) + (7z+10)=180\). This equation can be solved for \(z\):

\(\begin{align*} z + \dfrac12z + 7z+10&=180 \\ \dfrac{17}2z+10&=180 \\ 17z+20 &= 360 \\ 17z &= 340 \\ z &= 20\end{align*} \)

We can use the value \(z=20\) to determine the measure of the angles. 

Since \(\angle R = z^\circ\), \(\angle R=20^\circ\).

\(\begin{align*} \angle S &= \left(\dfrac12z \right)^\circ \\ &= \left(\dfrac12(20) \right)^\circ \\ &= 10 ^\circ \end{align*} \)

and 

\(\begin{align*} \angle T &= (7z+10)^\circ \\ &= (7(20)+10)^\circ \\ &= 150^\circ\end{align*} \)

Since one of the interior angles of the triangle (\(\angle T\)) is greater than \(90^\circ\), \(\triangle RST\) is an obtuse triangle. 

Check Your Understanding 2

Question — Version 1

\(\triangle JKL\) has interior angles \(\angle J = z^\circ\), \(\angle K = \left( \dfrac{1}{4}z \right)^\circ\), and \(\angle L = \left( \dfrac{5}{4} z \right)^\circ\). What is the measure of \(\angle L\)?

Answer — Version 1

The measure of \(\angle L\) is \(90^\circ\).

Feedback — Version 1

Since the sum of the interior angles of a triangle is \(180^\circ\), we can solve the following equation for \(z\):

\(\begin{align*} z + \dfrac{1}{4}z + \dfrac{5}{4}z & = 180 \\ \dfrac{4}{4} z + \dfrac{1}{4}z + \dfrac{5}{4}z & = 180 \\ \dfrac{10}{4}z &= 180 \\ z &= 180 \cdot \dfrac{4}{10} \\ z &= 72 \end{align*}\)

Use the value of \(z\) to determine the measure of \(\angle L\):

\(\begin{align*} \angle L &= \dfrac{5}{4}z \\ &= \dfrac{5}{4}(72) \\ &= 90\end{align*}\)

Therefore, the measure of \(\angle L\) is \(90^\circ\).

Question — Version 2

\(\triangle PQR\) has interior angles \(\angle P = x^\circ\), \(\angle Q = (2x-10)^\circ\), and \(\angle R = (4x+15)^\circ\). What is the measure of \(\angle R\)?

Answer — Version 2

The measure of \(\angle R\) is \(115^\circ\).

Feedback — Version 2

Since the interior angles of a triangle have a sum of \(180^\circ\), we can solve the following equation for \(x\):

\(\begin{align*} x + 2x - 10 + 4x + 15 &= 180 \\ 7x + 5 &= 180 \\ 7x & = 175 \\ x &= 25 \end{align*}\)

Use the value of \(x\) to determine the measure of \(\angle R\):

\(\begin{align*} \angle R &= (4x+15)^\circ \\ &=(4(25)+15)^\circ \\ &= 115^\circ \end{align*}\)

Therefore, the measure of \(\angle R\) is \(115^\circ\).

Conjectures and Counterexamples

In geometry, we often make conjectures. A conjecture is a statement that has not been proven to be true, but for which there is some evidence to suggest that it might be true. It is usually much easier to demonstrate that a conjecture is false than it is to prove that a conjecture is true. 

For instance, in a previous example we found an obtuse triangle containing a \(10^\circ\) angle. This may lead us to make the conjecture that all triangles that contain a \(10^\circ\) angle are obtuse. We will explore this statement further in the next example.  

Example 4

Solution

In a previous question, we saw that it is possible for a triangle to have interior angles of \(10^\circ\), \(20^\circ\), and \(150^\circ\). 

This triangle is obtuse, and so it is an example that supports the conjecture.

However, it is also possible to have a triangle with interior angles of \(10^\circ\), \(85^\circ\) and \(85^\circ\). This is because \(10^\circ+85^\circ+85^\circ=180^\circ\). 

This triangle is acute (since all of the angles are less than \(90^\circ\)), so we have found a counterexample.

This is enough to demonstrate that the conjecture is false. 

Example 5

Solution

Let's start by examining some examples of right-angled triangles. 

Triangle 1: 

It is possible to have a right-angled triangle where the other two angles are \(70^\circ\) and \(20^\circ\) 

(since \(90^\circ+70^\circ+20^\circ=180^\circ\)).  

\(70^\circ + 20 ^\circ = 90^\circ\), so \(70^\circ\) and \(20^\circ\) are complementary. 

This example supports the conjecture.

Triangle 2: 

It is also possible to have a right-angled triangle where the other two angles are \(45^\circ\) and \(45^\circ\) (since \(90^\circ+45^\circ+45^\circ=180^\circ\)). 

\(45^\circ\) and \(45^\circ\) are also complementary. 

This example supports the conjecture as well.

Although our supporting examples are not enough to prove the conjecture, they have helped us to see why it might be true. Let's see if we can prove it in general.

Proof:

Consider the right triangle \(\triangle ABC\). 

Since the interior angles of a triangle add to \(180^\circ\), \(\angle A + \angle B + \angle C = 180^\circ\). 

However, \(\angle A=90^\circ\), so \( 90^\circ + \angle B + \angle C = 180^\circ\).

It follows that \(\)\(\angle B + \angle C = 90^\circ\), and so \(\angle B\) and \(\angle C\) are complementary. 

Therefore, the statement is true.

Exterior Angles of a Triangle

We've spent some time talking about the interior angles of a triangle, now let's discuss the exterior angles. 

Identifying Exterior Angles

In the following triangle, one of the sides at each vertex has been extended producing a new angle at each vertex, these are the exterior angles.

It should be noted that there are actually two ways to create the exterior angles at each vertex since at each vertex, there are two sides that can be extended. 

If we extend the second side at each vertex, we produce even more angles. In fact, there are now two more angles at each vertex.

However, one of these new angles is opposite the nearest interior angle so it's equal to that interior angle.

The other new angle at each vertex is opposite to the first exterior angle that we found. So it's equal to that exterior angle.

This means that it doesn't matter which side of the triangle you extend. As long as you extend one side at each vertex, you will produce the correct exterior angles.

Relating Exterior and Interior Angles

Let's return to the triangle with only one exterior angle at each vertex.

You may have noticed that each interior angle is adjacent to and shares a vertex with an exterior angle. And each pair of interior and exterior angles forms a straight line. 

This means that the measures of each exterior angle and the adjacent interior angle add to \(180^\circ\).

We can say this in another way.

Thus, we can see three pairs of supplementary angles on this diagram.

Let's look at a numerical example of this.

Example 6

Solution

Although it's possible to answer this question without a diagram, drawing one may help us visualize a solution a little bit better. Let's sketch a triangle and mark the interior angles as \(98^\circ\), \(55^\circ\), and \(27^\circ\).

We can even add the exterior angles to this diagram by extending one of the sides at each vertex. 

The \(98^\circ\) interior angle is supplementary to the exterior angle right beside it. So that angle must be \(180^\circ-98^\circ=82^\circ\). 

Similarly, the exterior angle adjacent to the \(55^\circ\) interior angle is equal to \(180^\circ-55^\circ=125^\circ\).

The exterior angle next to the \(27^\circ\) interior angle is calculated the same way to obtain a measure of \(180^\circ-27^\circ=153^\circ\).

Therefore, the exterior angles are \(82^\circ\), \(125^\circ\), and \(153^\circ\).

Next, you'll have a chance to explore more properties of exterior angles.

Explore This 2

Description

In the following investigation, what is the relationship between the exterior angles in a triangle? Do you notice something regardless of the size, and/or shape of the triangle that you create? Also notice what happens to the sum of the exterior angles.

The following are examples of different triangles you could have created. 

Example 1

Sum of exterior angles: \(121^\circ + 125^\circ + 114^\circ= 360^\circ\)

Example 2

Sum of exterior angles: \(57^\circ + 156^\circ + 147^\circ= 360^\circ\)

Example 3

Sum of exterior angles: \(127^\circ + 90^\circ + 143^\circ= 360^\circ\)

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Interactive Version

Exterior Angles in a Triangle

Explore This 2 Summary

In the Explore This activity, you hopefully noticed an important fact about the exterior angles of a triangle. 

Illustrating the Sum of Exterior Angles

This demonstration might help you to see why this is the case.

Imagine that there is a triangular pathway and you're going to walk around it one time and end up facing the same way you started. 

Let's say you start at one vertex facing towards the next vertex.

To walk around the path, you need to walk along one side of the triangle.

Then, once you reach the second vertex, you need to rotate just enough so that you are pointed in the direction of the next vertex. The amount of rotation is the measure of the exterior angle of the triangle at that vertex.

Then, you would walk along the next side.

Once you reach the third vertex, you would rotate so that you're facing the direction of the starting vertex, which is another exterior angle. 

Finally, once you walk along the last side of the triangle and return to your starting position,

you would rotate enough so that you end up facing in the same direction you started.

Source: Man - kowalska-art/iStock/Getty Images

Ignoring how far you would have had to walk, think of how many full turns you would have completed to end up facing in the same direction you started. It would be just one full turn, which is \(360^\circ\).

Check Your Understanding 3

Question

\(\triangle ABC\) has two known exterior angles. The exterior angle at vertex \(B\) is \(100^\circ\) and at vertex \(C\) is \(165^\circ\). What is the measure of the missing exterior angle at vertex \(A\)?

Answer

The measure of the unknown exterior angle is \(95^\circ\).

Feedback

Let \(x^\circ\) represent the measure of the unknown exterior angle.

The sum of the exterior angles of a triangle is \(360^\circ\), so we can solve the following equation:

\(\begin{align*} x + 165 + 100 &= 360 \\ x+265 &= 360 \\ x &= 95 \end{align*}\)

Therefore, the missing exterior angle measures \(95^\circ\).

Note: This can also be determined by subtracting the two given angles from \(360^\circ\), i.e., \(360^\circ - 165^\circ - 100^\circ = 95^\circ\).

Justifying the Sum of the Exterior Angles of a Triangle

Let's consider why the sum of the exterior angles of a triangle is \(360^\circ\). 

Let the measures of the exterior angles be represented by \(d^\circ\), \(e^\circ\), and \(f^\circ\), as indicated in this diagram.

Remember that pairs of adjacent interior and exterior angles are supplementary. 

So the interior angle adjacent to \(d\) measures \((180-d)^\circ\).

Similarly, the interior angle adjacent to \(e\) measures \((180-e)^\circ\) and the interior angle adjacent to \(f\) measures \((180-f)^\circ\).

Since the sum of the interior angles is \(180^\circ\), we can write that \((180-d)+(180-e)+(180-f)=180\).

Let's simplify and rearrange this equation:

\(\begin{align*} 180-d+180-e+180-f &= 180 \\ 540 - d - e -f &= 180 \\ - d- e - f &= -360 \\ \end{align*} \)

Remember that we can divide both sides of the equation by \(-1\), so \(d+e+f=360\).

Therefore, the exterior angles of a triangle have a sum of \(360^\circ\), as required. 

Example 7 

Solution

The triangle can be drawn like this:

Since the exterior angles of a triangle add to \(360^\circ\), we can solve the following equation:

\(\begin{align*} x+x-20+x+5&=360 \\ 3x-15 &= 360 \\ 3x &= 375 \\ x &= 125\end{align*}\)

Example 8

Solution

Let \(p\) represent the measure of the exterior angle at vertex \(B\). 

Since the exterior angles of a triangle add to \(360^\circ\), 

\(\begin{align*} p &= 360^\circ - 110^\circ-95^\circ \\ p &= 155^\circ \end{align*} \)

Adjacent interior and exterior angles are supplementary, so \(p+q=180^\circ\). 

This means that \(q =25 ^\circ\). 

Alternate Method:

Since adjacent interior and exterior angles are supplementary, 

• The interior angle at vertex \(C\) is \(70^\circ\)
• The interior angle at vertex \(A\) is \(85^\circ\)

The interior angles of a triangle add to \(180^\circ\), so \( q = 180^\circ-70^\circ-85^\circ\), or \(q=25^\circ\).

Example 9

Solution

Let \(x\) represent the measure of the remaining interior angle (adjacent to \(y\)). 

Since the interior angles of a triangle add to \(180^\circ\), \(\)\( x = 180^\circ - 15^\circ- 21^\circ \) and so \(x=144^\circ\). 

Adjacent interior and exterior angles are supplementary, so \(x+y=180^\circ\).

This means that \(y=36^\circ\). 

It is interesting to note that \(36^\circ=15^\circ+21^\circ\). In fact, this is a pattern that can be observed on all triangles.

Exterior Angle Theorem

Here we're going to discuss what is known as the exterior angle theorem, which is related to what we observed in a previous example.

Let's see how this looks on a triangle. Consider this triangle where \(a^\circ\), \(b^\circ\), and \(c^\circ\) are the interior angles and \(d^\circ\), \(e^\circ\), and \(f^\circ\) are the exterior angles.

We'll start with the exterior \(\angle d\). The theorem states that the measure \(d\) is equal to the sum of the two interior angles at the other two vertices. These two angles are \(b\) and \(c\). Therefore, \(d=b+c\). 

Similarly, we can say that ‌\(e=a+c\),

and \(f=a+b\).

In summary, in this triangle, 

• \(d=b+c\)
• \(e=a+c\)
• \(f=a+b\)

Proof of Exterior Angle Theorem

Let's prove why this is the case. It is enough to show that the relationship holds for one exterior angle.

Let's look at an exterior angle that measures \(d^\circ\) and the non-adjacent, or remote, interior angles that measure \(b^\circ\) and \(c^\circ\).

Recall that the interior angle adjacent to \(d\) is supplementary to \(d\). So the interior angle adjacent to \(d\) has a measure of \((180-d)^\circ\).

We now have measures for all three of the interior angles of the triangle. Since the interior angles of the triangle have a sum of \(180^\circ\), we can say that

\( (180-d)+b+c =180 \)

Removing the brackets and subtracting \(180\) form both sides, we get that

\(\begin{align*} (180-d)+b+c &=180\\ 180-d + b + c &\;= 180 \\ b + c - d &\;= 0 \\ b + c &\;= d \end{align*}\)

Therefore, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices, \(b\) and \(c\), as required. 

Now let's use this result in an example.

Example 10

Solution

We can use the exterior angle theorem here since the angles \(k^\circ\) and \(38^\circ\) are interior angles that are not adjacent to the exterior angle marked \(108^\circ\). 

Using the Exterior Angle theorem, 

\(\begin{align*} k+38 &= 108 \\ k &\;= 70 \end{align*}\)

It is also possible to solve this and other way. 

Alternate Solution:

We could find the measure of the interior angle adjacent to \(108^\circ\), which is \(72^\circ\). The three interior angles of the triangle add to \(180^\circ\). So

\(\begin{align*} 72+38+k &= 180 \\ k &\;= 70 \end{align*}\)

Therefore, the interior angle at the third vertex measures \(70^\circ\).

Check Your Understanding 4

Question — Version 1

\(\triangle ABC\) has an interior angle at vertex \(B\) that measures \(35^\circ\), while an exterior angle at vertex \(C\) measures \(97^\circ\), as shown in the following diagram. Determine the measure of the exterior angle at vertex \(A\), indicated by \(x^\circ\).

Answer — Version 1

The value of \(x^\circ\) is \(118^\circ\).

Feedback — Version 1

The exterior angle adjacent to the \(35^\circ\) interior angle at vertex \(B\) measures \(145^\circ\), since those two angles are supplementary. 

The exterior angles of the triangle add to \(360^\circ\), so \(x+145+97=360\). It follows that \(x=118\).

Question — Version 2

\(\triangle ABC\) has an interior angle at vertex \(B\) that measures \(51^\circ\), while an exterior angle at vertex \(C\) measures \(102^\circ\), as shown in the following diagram. Determine the measure of the interior angle at vertex \(A\), indicated by \(x^\circ\).

Answer — Version 2

The value of \(x^\circ\) is \(51^\circ\).

Feedback — Version 2

The interior angle adjacent to the \(102^\circ\) interior angle at vertex \(C\) measures \(78^\circ\), since those two angles are supplementary. 

The interior angles of the triangle add to \(180^\circ\), so \(x+78+51=180\). It follows that \(x=51\).

Question — Version 3

\(\triangle ABC\) has interior angles at vertex \(B\) and \(C\) that measure \(35^\circ\) and \(18^\circ\) respectively, as shown in the following diagram. Determine the measure of the exterior angle at vertex \(A\), indicated by \(x^\circ\).

Answer — Version 3

The value of \(x^\circ\) is \(53^\circ\).

Feedback — Version 3

The measure of an exterior angle is equal to the sum of the interior angles at the other two vertices. So, \(x=18+35=53\).

Therefore, the value of \(x\) is \(53\).

Example 11

Solution

We begin by examining some triangles that have an exterior angle of \(100^\circ\).  

Triangle 1: 

A triangle with an exterior angle of \(100^\circ\) has an interior angle of \(80^\circ\) (since adjacent interior and exterior angles are supplementary).

By the Exterior Angle Theorem (or since all of the interior angles need to add to \(180^\circ\)), the other two interior angles add to \(100^\circ\). 

So, a triangle with an exterior angle of \(100^\circ\) could have interior angles of \(80^\circ\), \(95^\circ\) and \(5^\circ\).

This triangle is an obtuse triangle, since it has an interior angle greater than \(90^\circ\).

This example supports the conjecture. Recall, however, that this does not mean that the statement is true.  

Triangle 2: 

A triangle with an exterior angle of \(100^\circ\) could also have interior angles of \(80^\circ\), \(60^\circ\) and \(40^\circ\). 

This an acute triangle, not an obtuse triangle.

We have found a counterexample, and so the statement is false. 

Exterior Angles and Acute, Right, and Obtuse Triangles

Here are some notes on the exterior angles of different types of triangles:

• The interior angles of an acute triangle are all less than \(90^\circ\).
  • It follows that the exterior angles of an acute triangle are all obtuse (i.e., greater than \(90^\circ\)). 
• One of the interior angles of a right triangle is equal to \(90^\circ\), and the two other interior angles are acute.
  • It follows that one of the exterior angles of a right triangle is \(90^\circ\) and the other two exterior angles are obtuse. 
• One of the interior angles of an obtuse triangle is greater than \(90^\circ\), and the two other interior angles are acute.
  • It follows that one of the exterior angles of an obtuse triangle is acute and the other two exterior angles are obtuse.

Example 12

Solution

Both \(157^\circ\) and \(113^\circ\) are obtuse. However, all triangles have at least two obtuse exterior angles. We need to find the third exterior angle.

The third exterior angle is \(360^\circ-157^\circ-113^\circ=90^\circ\).

Therefore, the triangle is a right triangle.

Note: You can verify this result by determining the interior angles of the triangle. 

Relating the Sides and Angles of a Triangle

In addition to the relationships between the interior and exterior angles of a triangle, there is also a relationship between the interior angles and sides of a triangle.

In this triangle, the largest side, \(20\) cm, is opposite to the largest interior angle, \(100^\circ\).

In this triangle, the smallest side, \(3.5\) units, is opposite to the smallest interior angle, \(35^\circ\).

This relationship leads to the following fact:

Note that, since the exterior angles of a triangle are supplementary to the adjacent interior angles, a triangle that has two or more equal interior angles will have the same number of equal exterior angles. 

Equilateral Triangles

This means that all of the interior angles of an equilateral triangle are equal.

Since the interior angles of a triangle must add to \(180^\circ\), the interior angles of an equilateral triangle are all \(60^\circ\) (since \(3(60^\circ)=180^\circ\)). 

It also follows that all of the exterior angles of an equilateral triangle are equal.

Since the exterior angles of a triangle must add to \(360^\circ\), the exterior angles of an equilateral triangle are all \(120^\circ\) (since \(3(120^\circ)=360^\circ\)). This is also consistent with the fact that adjacent interior and exterior angles are supplementary (as \(60^\circ+120^\circ=180^\circ\)).

To recap:

Isosceles Triangles

Since an isosceles triangle has two sides that are equal in length, it must also have two interior angles that are equal in measure (and therefore two exterior angles that are equal in measure). 

Example 13

Solution

It may help to draw this triangle.

Since \(\angle P\) and \(\angle R\) are equal in measure, the sides opposite these angles will be equal in length. 

Therefore, sides \(QR\) and \(PQ\) are equal in length. 

Example 14

Solution

The equal sides of the triangle are indicated in the diagram. The angles opposite these equals sides are equal in measure, so there are two angles in the triangle with a measure of \(d^\circ\).

Since the interior angles of a triangle add to \(180^\circ\), we can solve the following equation to determine the value of \(d\)

\(\begin{align*} d+d+57&=180\\ 2d + 57 &= 180 \\ 2d &= 123 \\ d &= 61.5 \end{align*}\)

Therefore, the required angle measure is \(61.5^\circ\).

Example 15

Solution

There are a few different ways to solve this problem. Here are two of the methods.

Method 1

An isosceles triangle has two equal exterior angles, opposite the two equal sides.

Since a triangle's exterior angles must have a sum of \(360^\circ\):

\(\begin{align*} p &= 360 ^\circ - 105 ^\circ - 105^\circ\\ p &= 150 ^\circ \end{align*}\)

Method 2

Since the triangle is isosceles, there are two equal interior angles. 

These interior angles are supplementary to \(105^\circ\), and are equal to \(75^\circ\).

The Exterior Angle theorem states an exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices.

So, \(p=75^\circ+75^\circ\), or \(p=150^\circ\). 

Next, let's practice using the angle properties of isosceles and other triangles on some more complicated figures. 

Check Your Understanding 5

Question — Version 1

\(\triangle ABC\) is an isosceles triangle that with \(\angle ABC\) measuring \(36^\circ\), as shown in the following diagram. Determine the measure of \(\angle BCA\), indicated by \(x^\circ\).

Answer — Version 1

The value of \(x\) is \(108\).

Feedback — Version 1

The angles opposite the equal sides both measure \(36^\circ\).

Since the interior angles of a triangle have a sum of \(180^\circ\), \(x+36+36=180\).

This can be solved to obtain \(x=108\).

Question — Version 2

\(\triangle ABC\) is an isosceles triangle that with \(\angle ACB\) measuring \(28^\circ\), as shown in the following diagram. Determine the measure of \(\angle BAC\), indicated by \(x^\circ\).

Answer — Version 2

The value of \(x\) is \(76\).

Feedback — Version 2

The angles oppsite the equal sides both have a measure of \(x^\circ\).

Since the interior angles of a triangle have a sum of \(180^\circ\), \(28+x+x=180\).

This equation can be simplified and solved:

\(\begin{align*} 28 + 2x &= 180 \\ 2x &= 152 \\ x &= 76\end{align*}\)

Therefore, the value of \(x\) is \(76\).

Example 16

For both of these figures it is impossible to identify \(x\) directly. We are going to need to find the measures of some additional angles first.

Solution — Figure A

Let's start with Figure A. 

Since we can't find \(x\) directly, we need to look for other angles that we can find more easily. Notice the \(\triangle CDE\) has two equal sides, and so it's isosceles.

The angles opposite the two equal sides will be equal. So \(\angle CED=46^\circ\).

If we continue to look at \(\triangle CDE\), we know that its interior angles must add to \(180^\circ\). So the missing angle, \(\angle DCE\), will be 

\(\begin{align*} \angle DCE+46^\circ+46^\circ &= 180^\circ \\ \angle DCE &\;= 88^\circ \end{align*}\)

Now, \(\angle ACB\) and \(\angle DCE\) are opposite angles, so they are equal. This means that \(\angle ACB\) measures \(88^\circ\) as well.

We finally have enough information to determine \(x\) in \(\triangle ABC\).

The interior angles of \(\triangle ABC\) also add to \(180^\circ\). So 

\(\begin{align*} x+15+88 &= 180 \\ x &\;=77 \end{align*}\)

Therefore, the angle measure indicated by \(x\) is \(77^\circ\).

Solution — Figure B

For Figure B, we also need to find the measures of other angles before finding the value of \(x\).

We can see that \(\angle QRT\) and \(\angle SRT\) share a vertex and form a straight line. So they're supplementary. 

This means that these two angles must add to \(180^\circ\). So, \(\angle SRT\) measures \(110^\circ\), since \(\angle QRT\) measures \(70^\circ\).

Now let's look at \(\triangle RST\). 

The interior angles of \(\triangle RST\) must add to \(180^\circ\). So

\(\begin{align*} \angle RST+10^\circ+110^\circ &= 180^\circ \\ \angle RST &\;= 60^\circ \end{align*}\)

Consider the larger \(\triangle QST\). It has two equal sides, so it is isosceles.

This means that two of its angles are equal, \(\angle RST\) and \(\angle RQT\). Thus, \(\angle RQT\) measures \(60^\circ\) as well.

We can finally consider \(\triangle QRT\).

The interior angles of \(\triangle QRT\) also add to \(180^\circ\). So

\(\begin{align*} x+60+70 &= 180 \\ x &\;= 50 \end{align*}\)

Therefore, the angle indicated by \(x\) measures \(50^\circ\).

Let's take a look at the larger \(\triangle QST\) in Figure B.

For \(\triangle QST\), the interior \(\angle T\) is

\(\begin{align*} \angle T &= 50^\circ+10^\circ \\ &= 60^\circ \end{align*}\)

The interior angles of \(\triangle QST\) are all \(60^\circ\).

It turns out that \(\triangle QST\) is an equilateral triangle, although we didn't need to know that to solve the question.

Using Angles to Classify Triangles

Because of the relationship between the side lengths and interior angles of a triangle, it is possible to fully classify any triangle using its interior angles alone (without knowing any side lengths).

Here are the different types of triangles:

Type of Triangle	Interior Angle Properties:	Example
Acute equilateral	All angles are equal and less than \(90^\circ\). (In fact, they are all \(60^\circ\).)
Acute isosceles	Two angles are equal, and all angles are less than \(90^\circ\).
Acute scalene	No angles are equal, and all angles are less than \(90^\circ\).
Right isosceles	Two angles are equal (both \(45^\circ\)), and one angle equals \(90^\circ\).
Right scalene	No angles are equal, and one angle equals \(90^\circ\).
Obtuse isosceles	Two angles are equal, and one angle is greater than \(90^\circ\).
Obtuse scalene	No angles are equal, and one angle is greater than \(90^\circ\).

Note that it is not possible to have a right equilateral or an obtuse equilateral triangle, since an equilateral triangle must have all three interior angles equal to \(60^\circ\). 

Example 17

Solution

We can classify \(\triangle DEF\) as equilateral, isosceles, or scalene if we know the measure of each of the interior angles. To do this, we need to determine the value of \(y\). 

The sum of the interior angles of a triangle is \(180^\circ\), so we can solve the following equation:

\(\begin{align*} 3y+\dfrac12y+6y-10& =180\\ \dfrac{19}2y - 10 &= 180 \\ \dfrac{19}2y &= 190 \\ 19y &= 380 \\ y &= 20 \end{align*} \)

Now, we can use the value of \(y\) to determine the measures of the interior angles.

The interior angles of \(\triangle DEF\) are \(60^\circ \), \(10^\circ\), and \(110^\circ\). Since they all have different measures, the triangle is scalene. 

Try This Revisited

Solution

There are two cases to consider.

Case 1: The two equal angles are larger than the third angle (more specifically, twice as large). 

Case 2: The two equal angles are smaller than the third angle (which would be twice their size).

We can solve each of these cases separately.

Case 1: The two equal angles are larger than the third angle.

Let \(x^\circ\) represent the measure of the third angle. Then, the two equal angles each have a measure of \((2x)^\circ\).

We know that \(x+2x+2x=180\). Simplifying and solving, we have that \(5x=180\), or \(x=36\). 

That means that the triangle contains one angle measuring \(36^\circ\) and two angles measuring \(72^\circ\). 

All of the angles in this triangle are less than \(90^\circ\), so this triangle is acute.

Case 2: The two equal angles are smaller than the third angle.

Let \(y^\circ\) represent the measure of each of the equal angles. Then, the third angle has a measure of \((2y)^\circ\). 

We know that \(y+y+2y=180\). Simplifying and solving this equation gives \(4y=180\) or \(y=45\). 

This means that the triangle contains two angles measuring \(45^\circ\) and one angle measuring \(90^\circ\).

This triangle contains one \(90^\circ\) angle, so this triangle is right.

Therefore, an isosceles triangle where one angle measure is twice the other could be an acute triangle or a right triangle.

Wrap-Up

Take It With You

Consider the quadrilaterals \(ABCD\) and \(PQRS\), pictured here.

What is the sum of the interior angles in each of these quadrilaterals? How do you know?  

Extension: The Triangle Inequality

In this lesson, we discussed angle relationships in triangles. We also discussed the relationship between the side lengths and angles in a triangle.

There are also important relationships between the side lengths of a triangle. One of these relationships is the Pythagorean Theorem, which you may have seen in other lessons. The Pythagorean Theorem relates the side lengths of a right-angled triangle.

Another important relationship that applies to all triangles is known as the triangle inequality. 

For both of the triangles displayed above, \(a \lt b+c\), \(b \lt a+c\), and \(c \lt a+b\).

If one of the side lengths is longer than the sum of the other two, those two sides won't meet. This does not produce a triangle.

If \(a=b+c\), \(b=a+c\) or \(c=a+b\), this produces what is known as a degenerate triangle.

Consider a triangle with side lengths of \(2\), \(5\), and \(7\) units. 

Although \(2 \lt 5+7\), and \(5 \lt 2+7\), we have that \(7=2+5\).

This produces a degenerate triangle, as seen below.

In geometry, we are most often interested in non-degenerate triangles (i.e., triangles that have an area greater than \(0\)). 

Example 18

Solution — Part A

Since \(17 \not \le 8+8\), the triangle inequality does not hold for these three side lengths.

(This is the case even though \(8 \lt 8+17\), since all three of the inequalities must hold.)

Therefore, it is not possible to create any triangle (even a degenerate one) with side lengths of \(8\), \(8\), and \(17\) units.

Solution — Part B

• \(10 \lt 3+8\)
• \(3 \lt 8 + 10\), and 
• \(8 \lt 3+10\)

Therefore, it is possible to create a non-degenerate triangle with side lengths of \(3\), \(8\), and \(10\) units. 

Solution — Part C

• \(2 \lt 3+5\)
• \(3 \lt 2+5\), and
• \(5=2+3\).

Thus, the triangle inequality holds for these three side lengths. 

However, \(5 \not \lt 2+3\), so we would have a degenerate triangle. 

Therefore, it is not possible to create a non-degenerate triangle with side lengths of \(2\), \(3\), and \(5\) units.

Note: Although all three inequalities in the triangle inequality must hold, we only need to check one of them. Here's why:

Consider three non-negative numbers \(a\), \(b\), and \(c\), where none of the numbers are greater than \(c\). That is, \(a \le c\) and \(b \le c.\)

Then, no matter what the three numbers are, it must be true that \(a \le b+c\) (since \(b \ge 0\)) and that \(b \le a+c \) (since \(a \ge 0\)). 

The only inequality we can't be sure about is whether or not \(c \le a+b\).

Since the side lengths of a triangle are always non-negative, we can apply this idea to the triangle inequality.