Explainer: Inequality in One Triangle: Angle-Side

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 𝐷𝐸𝐹, 𝐷𝐸>𝐸𝐹, then π‘šβˆ πΉπ‘šβˆ π·.

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, π‘šβˆ πΉ>π‘šβˆ π·.

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

Given that 𝐴𝐡=92centimeters, 𝐴𝐢=91centimeters, and 𝐢𝐸=𝐡𝐷, choose the correct relationship between π‘šβˆ π΄πΈπ· and π‘šβˆ π΄π·πΈ.

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 𝐴𝐡=92centimeters and 𝐴𝐢=91centimeters, so 𝐴𝐢<𝐴𝐡 and 𝐢𝐸=𝐡𝐷. Also, 𝐴𝐷=π΄π΅βˆ’π΅π· and 𝐴𝐸=π΄πΆβˆ’πΆπΈ; that is, 𝐴𝐸=π΄πΆβˆ’π΅π·. As 𝐴𝐢<𝐴𝐡, 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 π‘šβˆ π΄πΈπ·>π‘šβˆ π΄π·πΈ.

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

Which is the correct relationship between 𝐴𝐢 and 𝐴𝐡?

Answer

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

∠𝐷𝐴𝐸 and ∠𝐡 are corresponding angles formed by a line, ⃖⃗𝐷𝐡, intersecting two parallel lines, ⃖⃗𝐴𝐸 and ⃖⃗𝐡𝐢. So, they are equal in measure: π‘šβˆ π΅=π‘šβˆ π·π΄πΈ=67degrees.

∠𝐸𝐴𝐢 and ∠𝐢 are alternate interior angles formed by a line intersecting two parallel lines. So, they are equal in measure: π‘šβˆ πΆ=π‘šβˆ πΈπ΄πΆ=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 π‘šβˆ πΆ<π‘šβˆ π΅, we conclude that 𝐴𝐢>𝐴𝐡.

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

Choose the correct relationship between 𝐴𝐷 and 𝐴𝐡.

  1. 𝐴𝐷<𝐴𝐡
  2. 𝐴𝐷=𝐴𝐡
  3. 𝐴𝐷>𝐴𝐡

Answer

𝐴𝐷 and 𝐴𝐡 are two sides of triangle 𝐴𝐡𝐷. 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 𝐴𝐷𝐡 and 𝐴𝐡𝐷.

We observe that triangle 𝐴𝐡𝐹 is an isosceles triangle; therefore, π‘šβˆ π΅π΄πΉ=π‘šβˆ π΄π΅πΉ. Hence, π‘šβˆ π΄π΅πΉ=38∘. Triangle 𝐴𝐹𝐷 is also an isosceles triangle; therefore, π‘šβˆ πΉπ΄π·=π‘šβˆ πΉπ·π΄. We can find the measures of these angles by writing that the sum of angles in triangle 𝐴𝐡𝐷 is 180∘: π‘šβˆ π΄π΅π·+π‘šβˆ π΅π·π΄+π‘šβˆ π·π΄π΅=180∘.

And with π‘šβˆ π΅π·π΄=π‘šβˆ πΉπ·π΄, π‘šβˆ π·π΄π΅=π‘šβˆ πΉπ΄π·+38∘, π‘šβˆ πΉπ΄π·=π‘šβˆ πΉπ·π΄, and π‘šβˆ π΄π΅π·=38∘, we get 38∘+π‘šβˆ πΉπ·π΄+π‘šβˆ πΉπ·π΄+38∘=180∘, which simplifies to 2π‘šβˆ πΉπ·π΄=180βˆ’2Γ—38; that is, π‘šβˆ πΉπ·π΄=52∘.

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

Now, we can write that π‘šβˆ π΄π΅π·=38∘ and π‘šβˆ π΅π·π΄=π‘šβˆ πΉπ·π΄=52∘.

Therefore, we have π‘šβˆ π΄π΅π·<π‘šβˆ π΅π·π΄. 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<𝐴𝐡.

Example 5: Applying the Angle-Side Inequality

Determine the correct inequality in the following figure.

  1. 𝐴𝐡<𝐢𝐡
  2. 𝐴𝐡>𝐴𝐢
  3. 𝐴𝐡>𝐢𝐡
  4. 𝐴𝐢<𝐢𝐡

Answer

In the figure, we observe that ⃖⃗𝐴𝐷 and ⃖⃗𝐡𝐢 are parallel lines. The angles 𝐷𝐴𝐡 and 𝐴𝐡𝐢 are alternate angles; therefore, their measures are equal. So, we have π‘šβˆ π΄π΅πΆ=66∘.

Writing that the sum of the angles in triangle 𝐴𝐡𝐢 is 180∘ gives us π‘šβˆ π΄π΅πΆ+π‘šβˆ π΅πΆπ΄+π‘šβˆ π΅π΄πΆ=180∘.

Substituting, we find that 66+π‘šβˆ π΅πΆπ΄+52=180∘; thus, we have π‘šβˆ π΅πΆπ΄=62∘.

Now, we can write the following inequality: π‘šβˆ π΅π΄πΆ<π‘šβˆ π΅πΆπ΄<π‘šβˆ π΄π΅πΆ.

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 𝐢𝐡<𝐴𝐡<𝐴𝐢; that is, 𝐴𝐢>𝐴𝐡>𝐢𝐡.

Among the options given in the question, only 𝐴𝐡>𝐢𝐡 is correct.

Key Points

  1. 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.
  2. It follows that if the lengths of the sides of a triangle are known, then the angles can be ordered according to their measures.
  3. 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.
  4. 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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