Lesson Explainer: Inequality in One Triangle: Angle-Side Mathematics

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: Using 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 𝐷𝐸𝐹, 𝐷𝐸>𝐸𝐹, then 𝑚𝐹𝑚𝐷.

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 𝐷𝐸>𝐸𝐹; therefore, 𝑚𝐹>𝑚𝐷.

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

Given that 𝐴𝐵=92centimetres, 𝐴𝐶=91centimetres, and 𝐶𝐸=𝐵𝐷, choose the correct relationship between 𝑚𝐴𝐸𝐷 and 𝑚𝐴𝐷𝐸.

Answer

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

We know that 𝐴𝐵=92centimetres and 𝐴𝐶=91centimetres, so 𝐴𝐶<𝐴𝐵 and 𝐶𝐸=𝐵𝐷. Also, 𝐴𝐷=𝐴𝐵𝐵𝐷 and 𝐴𝐸=𝐴𝐶𝐶𝐸; that is, 𝐴𝐸=𝐴𝐶𝐵𝐷. As 𝐴𝐶<𝐴𝐵, it follows that 𝐴𝐸<𝐴𝐷.

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. 𝐴𝐸𝐷 is opposite 𝐴𝐷 and 𝐴𝐷𝐸 is opposite 𝐴𝐸; therefore, we conclude that 𝑚𝐴𝐸𝐷>𝑚𝐴𝐷𝐸.

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

Which is the correct relationship between 𝐴𝐶 and 𝐴𝐵?

Answer

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

𝐷𝐴𝐸 and 𝐵 are corresponding angles formed by a line, 𝐷𝐵, intersecting two parallel lines, 𝐴𝐸 and 𝐵𝐶. So, they are equal in measure: 𝑚𝐵=𝑚𝐷𝐴𝐸=67degrees.

𝐸𝐴𝐶 and 𝐶 are alternate interior angles formed by a line intersecting two parallel lines. So, they are equal in measure: 𝑚𝐶=𝑚𝐸𝐴𝐶=61degrees.

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 𝑚𝐶<𝑚𝐵, we conclude that 𝐴𝐶>𝐴𝐵.

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

Choose the correct relationship between 𝐴𝐷 and 𝐴𝐵.

  1. 𝐴𝐷<𝐴𝐵
  2. 𝐴𝐷=𝐴𝐵
  3. 𝐴𝐷>𝐴𝐵

Answer

𝐴𝐷 and 𝐴𝐵 are two sides of triangle 𝐴𝐵𝐷. 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 𝐴𝐷𝐵 and 𝐴𝐵𝐷.

We observe that triangle 𝐴𝐵𝐹 is an isosceles triangle; therefore, 𝑚𝐵𝐴𝐹=𝑚𝐴𝐵𝐹. Hence, 𝑚𝐴𝐵𝐹=38. Triangle 𝐴𝐹𝐷 is also an isosceles triangle; therefore, 𝑚𝐹𝐴𝐷=𝑚𝐹𝐷𝐴. We can find the measures of these angles by writing that the sum of angles in triangle 𝐴𝐵𝐷 is 180: 𝑚𝐴𝐵𝐷+𝑚𝐵𝐷𝐴+𝑚𝐷𝐴𝐵=180.

And with 𝑚𝐵𝐷𝐴=𝑚𝐹𝐷𝐴, 𝑚𝐷𝐴𝐵=𝑚𝐹𝐴𝐷+38, 𝑚𝐹𝐴𝐷=𝑚𝐹𝐷𝐴, and 𝑚𝐴𝐵𝐷=38, we get 38+𝑚𝐹𝐷𝐴+𝑚𝐹𝐷𝐴+38=180, which simplifies to 2𝑚𝐹𝐷𝐴=1802×38; that is, 𝑚𝐹𝐷𝐴=52.

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

Now, we can write that 𝑚𝐴𝐵𝐷=38 and 𝑚𝐵𝐷𝐴=𝑚𝐹𝐷𝐴=52.

Therefore, we have 𝑚𝐴𝐵𝐷<𝑚𝐵𝐷𝐴. 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 𝐴𝐷<𝐴𝐵.

Example 5: Applying the Angle-Side Inequality

Determine the correct inequality in the following figure.

  1. 𝐴𝐵<𝐶𝐵
  2. 𝐴𝐵>𝐴𝐶
  3. 𝐴𝐵>𝐶𝐵
  4. 𝐴𝐶<𝐶𝐵

Answer

In the figure, we observe that 𝐴𝐷 and 𝐵𝐶 are parallel lines. The angles 𝐷𝐴𝐵 and 𝐴𝐵𝐶 are alternate angles; therefore, their measures are equal. So, we have 𝑚𝐴𝐵𝐶=66.

Writing that the sum of the angles in triangle 𝐴𝐵𝐶 is 180 gives us 𝑚𝐴𝐵𝐶+𝑚𝐵𝐶𝐴+𝑚𝐵𝐴𝐶=180.

Substituting, we find that 66+𝑚𝐵𝐶𝐴+52=180; thus, we have 𝑚𝐵𝐶𝐴=62.

Now, we can write the following inequality: 𝑚𝐵𝐴𝐶<𝑚𝐵𝐶𝐴<𝑚𝐴𝐵𝐶.

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 𝐶𝐵<𝐴𝐵<𝐴𝐶; that is, 𝐴𝐶>𝐴𝐵>𝐶𝐵.

Among the options given in the question, only 𝐴𝐵>𝐶𝐵 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.

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