In this explainer, we will learn how to form inequalities involving the measures of angles in a triangle given the lengths of the sides of the triangle.

In an isosceles triangle, we can recall that the angles opposite the sides of equal length are of equal measure. The converse to this statement is also true: if two angles in a triangle have an equal measure, then the sides opposite these angles have the same length. We can use this idea to consider what happens if we have two sides in a triangle with different lengths.

For example, consider the following triangle with two given side lengths and the measures of the opposite angles marked.

We can compare the measures of the angles by flipping the triangle so that the angles are in the same orientation.

We can see that the angle of measure is larger than the angle of measure . In other words, the angle opposite the longer side has the larger measure.

This result holds true in general. If we have a triangle , where , then it is always true that ; the angle opposite the longer s ide has a larger measure. We can state this result formally as follows.

### Theorem: Angle Comparison Theorem in Triangles

If we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure than the angle opposite the shorter side. In particular, consider .

If we know that , then .

It is also worth noting that we can apply this result with the inequality reversed. In particular, if we have a triangle where two sides have unequal lengths, then the angle opposite the shorter side has a smaller measure.

Letβs see an example of applying this result to construct an inequality for a triangle.

### Example 1: Finding the Relation between the Angles of a Triangle

Complete the following using <, =, or >: If in the triangle , , then .

### Answer

We recall that the angle comparison theorem in triangles tells us that if we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure than the angle opposite the shorter side.

Since we have a triangle and we are told that , we can conclude that the angle opposite must have a larger measure than th e angle opposite . We can sketch the triangle to help determine the angles opposite these sides.

We see that is opposite the side of length and is opposite the side of length . The angle opposite the longer side must have a larger measure.

Hence,

In our next example, we will see how to apply this result to form multiple inequalities for the measures of angles in a triangle when we are given all of its side lengths.

### Example 2: Forming Inequalities for the Measures of Angles in a Triangle

Consider this triangle.

Fill in the blanks in the following statements using =, <, or >.

### Answer

We first recall that the angle comparison theorem in triangles tells us that if we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure than the angle opposite the shorter side. Equivalently, the angle opposite the shorter side has a smaller measure than the angle opposite the longer side.

Since we are given all three side lengths in the triangle, we can compare any pair of side lengths to construct inequalities for the measures of the angles opposite the sides.

**Part 1**

We want to compare the measures of the angles at vertices and . The sides opposite these angles are and respectively. We know that and that . Thus,

The angle comparison theorem in triangles then tells us that the angle opposite must have a larger measure than the angle opposite .

Hence,

**Part 2**

We can follow the same process to compare the measures of the angles at vertices and . The sides opposite these angles are and respectively. We know that and that . Thus,

The angle comparison theorem in triangles then tells us that the angle opposite must have a larger measure than the angle opposite .

Hence,

**Part 3**

To compare the measures of the angles at vertices and , we consider the sides opposite these angles, and . We know that and that . Thus,

The angle comparison theorem in triangles then tells us that the angle opposite must have a smaller measure than the angle opposite .

Hence,

In the previous example, we saw that we can use the angle comparison theorem in triangles to construct three inequalities for the measures of the angles in a triangle by using its side lengths.

In particular, we can use this to order all of the measures of the angles in the triangle. For example, in the previous question, we saw that and that . This means that must be larger than . We can write this as a compound inequality:

This application of the angle comparison theorem gives us the following property.

### Property: Ordering the Angle Measures in a Triangle

If we have a triangle where all of its sides have unequal lengths, then the angle opposite the longest side has the largest measure and the angle opposite the shortest side has the shortest measure.

In particular, consider .

If we know that , then we can conclude that .

In our next example, we will compare the measures of two angles in a triangle using the angle comparison theorem and the axioms of inequalities.

### Example 3: Forming an Inequality for the Measures of Angles in a Triangle

Given , , and , choose the correct relationship between and .

### Answer

We are asked to compare the measures of and , two angles of triangle . In order to do this, we first recall that the angle comparison theorem in triangles tells us that if we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has the larger measure.

This means we can compare the measures of and by comparing the lengths of their respective opposite sides, and . We can add the lengths of and onto the diagram.

We are told that and , so

We can recall that subtracting the same value from both sides of the inequality will give us an equivalent inequality. We can also note that and and that . Thus, we can subtract from both sides of the inequality to get

We can now use the fact that to rewrite the left-hand side of the inequality:

Now, we can substitute and to find that

In a triangle, if one side is longer than another side, then the angle opposite the shorter side has a smaller measure than that opposite the longer side. is opposite and is opposite ; therefore, we conclude that .

Before we move on to our next example, it is worth noting that we can prove the angle comparison theorem in triangles using our knowledge of isosceles triangles, inequalities, and exterior angles in triangles.

First, letβs assume we have a triangle with .

We want to show that . Since , there must be a point on such that . We can add this point onto our diagram and then sketch .

Since , we note that is an isosceles triangle and hence . We can add these angles onto the diagram.

We can now note that is an exterior angle in . We recall that the measure of the exterior angle in a triangle is equal to the sum of the measures of the two opposite interior angles or, alternatively, use the fact that it is a supplementary angle to .

Thus,

In particular, we have

We must also have

Since , we have shown

In our next example, we will use the angle comparison theorem in triangles to prove a result from a figure.

### Example 4: Proving Angle Inequalities in a Triangle

In the figure, and . Which of the following is true?

### Answer

Since we are given the lengths of the sides in the figure and are asked to use this to compare the measures of the angles, we can recall the angle comparison theorem in triangles. This tells us that if we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure.

To apply this result, we need to be working with a triangle with at least two side lengths we can compare. We can do this by adding in the line segment .

In , we know that , so the angle opposite has a larger measure than the angle opposite . We have that

Similarly, in , we are told that , so the angle opposite has a larger measure than the angle opposite . We have that

By adding the measures of the angles to give the measures of the angles at vertices and , we have the following:

We know that and hence, is the combination of the two larger angles.

Hence,

In our final example, we will use the angle comparison theorem in triangles to prove a result about the lengths of medians in a triangle.

### Example 5: Identifying the Type of an Angle Using the Angle Comparison Theorem

Given that is a median of triangle and that , what type of angle is ?

- Obtuse
- Right
- Reflex
- Acute

### Answer

We can first recall the angle comparison theorem in triangles. This tells us that if we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure. We can apply this to the two triangles and .

First, since , the angle opposite must have a smaller measure than the angle opposite in . Thus,

Second, since , we also have that . Therefore, the angle opposite must have a smaller measure than the angle opposite in . Thus,

We can note that and combine to make , so the sum of their measures gives the measure of . We can add these two inequalities to get

Since , we can rewrite the right-hand side of this inequality to get

The right-hand side of this inequality is the sum of the measures of two of the interior angles of . We can add to both sides of the inequality to get

We know that the measures of the interior angles in a triangle sum to give . Therefore, we can rewrite the right-hand side of the inequality as and we can simplify the left-hand side to get

Finally, we divide both sides of the inequality by 2 to get

Hence, is acute.

Letβs finish by recapping some of the important points from this explainer.

### Key Points

- If we have a triangle where two sides have unequal lengths, then the angle opposite the longer side has a larger measure than the angle opposite the shorter side. Equivalently, the angle opposite the shorter side has a smaller measure than the angle opposite the longer side.
- If we have a triangle where all of its sides have unequal lengths, then the angle opposite the longest side has the largest measure and the angle opposite the shortest side has the shortest measure. This allows us to order the measures of the interior angles of a triangle.