In this explainer, we will learn how to identify the relationships between angles and side lengths in a triangle to deduce the inequality in one triangle.

We know that if two sides of a triangle have the same length, then the angles opposite those sides are of the same measure. We are going to see that, in any triangle, there exists a relationship between the sides and the angles in the form of an inequality.

### Angle-Side Inequality in Triangles

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. Inversely, if the measure of one angle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than the side opposite the greater angle.

### How to Use the Angle-Side Inequality in a Triangle

If the lengths of the sides of a triangle are known, then the angles can be ordered according to their measures.

Inversely, if the measures of the angles of a triangle are known, then the sides can be ordered according to their lengths.

### Example 1: Stating the Angle-Side Inequality in a Triangle

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

### Answer

According to the angle-side inequality in triangles, we know 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. Here, the side ; therefore, .

### Example 2: Using the Angle-Side Inequality in a Triangle to Compare Angles

Given that , , 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 need to find out how the lengths of their respective opposite sides, and , compare and then use the angle-side inequality in the triangle.

We know that and , so and . Also, and ; that is, . As , it follows 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 .

### Example 3: Using the Angle-Side Inequality in a Triangle to Compare Lengths

Which is the correct relationship between and ?

### Answer

We are asked to compare the lengths of and , two sides of triangle . In order to do this, we need to find out how the measures of the angles opposite and , that is, and , compare and then use the angle-side inequality in triangle .

and are corresponding angles formed by a line, , intersecting two parallel lines, and . So, they are equal in measure: .

and are alternate interior angles formed by a line intersecting two parallel lines. So, they are equal in measure: .

In a triangle, if the measure of one angle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than the side opposite the greater angle.

As , we conclude that .

### Example 4: Applying the Angle-Side Inequality to Compare Lengths

Choose the correct relationship between and .

### Answer

and are two sides of triangle . We can compare their lengths using the angle-side inequality in a triangle. In order to do this, we need to know the measures of the angles and .

We observe that triangle is an isosceles triangle; therefore, . Hence, . Triangle is also an isosceles triangle; therefore, . We can find the measures of these angles by writing that the sum of angles in triangle is :

And with , , , and , we get which simplifies to that is,

Note that we could have also identified that triangle is a right triangle at since its median is half of the side .

Now, we can write that and

Therefore, we have . The angle-side inequality states that, in a triangle, if the measure of one angle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than the side opposite the greater angle. So, we have

### Example 5: Applying the Angle-Side Inequality

Determine the correct inequality in the following figure.

### Answer

In the figure, we observe that and are parallel lines. The angles and are alternate angles; therefore, their measures are equal. So, we have

Writing that the sum of the angles in triangle is gives us

Substituting, we find that thus, we have

Now, we can write the following inequality:

The angle-side inequality states that, in a triangle, if the measure of one angle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than the side opposite the greater angle. So, we have that is,

Among the options given in the question, only is correct.

### Key Points

- 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.
- It follows that if the lengths of the sides of a triangle are known, then the angles can be ordered according to their measures.
- Inversely, if the measure of one angle in a triangle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than that of the side opposite the greater angle.
- It follows that if the measures of the angles in a triangle are known, then the sides can be ordered according to their lengths.