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5.4: Inequalities in Triangles

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Determine relationships among the angles and sides of a triangle.
  • Understand the Triangle Inequality Theorem.
  • Understand the SAS Inequality Theorem and its converse.

Review Queue

Solve the following inequalities.

  1. \begin{align*}4x - 9 \le 19\end{align*}
  2. \begin{align*}-5 > -2x + 13\end{align*}
  3. \begin{align*}\frac{2}{3}x + 1 \ge 13\end{align*}
  4. \begin{align*}-7 < 3x - 1 < 14\end{align*}

Know What? Two planes take off from LAX. Their flight patterns are to the right. Both planes travel 200 miles, but which one is further away from LAX?

Comparing Angles and Sides

Look at the triangle to the right. The sides of the triangle are given. Can you determine which angle is the largest? The largest angle will be opposite 18 because it is the longest side. Similarly, the smallest angle will be opposite 7, which is the shortest side.

Theorem 5-9: If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

Converse of Theorem 5-9: If one angle in a triangle is larger than another angle in a triangle, then the side opposite the larger angle will be longer than the side opposite the smaller angle.

To prove these theorems, we will do so indirectly. This will be done in the extension at the end of this chapter.

Example 1: List the sides in order, from shortest to longest.

Solution: First, find \begin{align*}m \angle A\end{align*}. From the Triangle Sum Theorem:

\begin{align*}m \angle A + 86^\circ + 27^\circ &= 180^\circ\\ m \angle A &= 67^\circ\end{align*}

\begin{align*}86^\circ\end{align*} is the largest angle, so \begin{align*}AC\end{align*} is the longest side. The next angle is \begin{align*}67^\circ\end{align*}, so \begin{align*}BC\end{align*} would be the next longest side. \begin{align*}27^\circ\end{align*} is the smallest angle, so \begin{align*}AB\end{align*} is the shortest side. In order, the answer is: \begin{align*}AB, \ BC, \ AC\end{align*}.

Example 2: List the angles in order, from largest to smallest.

Solution: Just like with the sides, the largest angle is opposite the longest side. The longest side is \begin{align*}BC\end{align*}, so the largest angle is \begin{align*}\angle A\end{align*}. Next would be \begin{align*}\angle B\end{align*} and then \begin{align*}\angle A\end{align*}.

Triangle Inequality Theorem

Can any three lengths make a triangle? The answer is no. For example, the lengths 1, 2, 3 cannot make a triangle because \begin{align*}1 + 2 = 3\end{align*}, so they would all lie on the same line. The lengths 4, 5, 10 also cannot make a triangle because \begin{align*}4 + 5 = 9\end{align*}.

The arc marks show that the two sides would never meet to form a triangle.

Triangle Inequality Theorem: Two sides must add up to be greater than the third side.

Example 3: Do the lengths below make a triangle?

a) 4.1, 3.5, 7.5

b) 4, 4, 8

c) 6, 7, 8

Solution: Even though the Triangle Inequality Theorem says “the sum of the length of any two sides,” add up the two shorter sides. They must be greater than the third.

a) \begin{align*}4.1 + 3.5 > 7.5\end{align*} Yes, \begin{align*}7.6 > 7.5\end{align*}

b) \begin{align*}4 + 4 = 8\end{align*} No, not a triangle. Two lengths cannot equal the third.

c) \begin{align*}6 + 7 > 8\end{align*} Yes, \begin{align*}13 > 8\end{align*}

Example 4: Find the length of the third side of a triangle if the other two sides are 10 and 6.

Solution: The Triangle Inequality Theorem can also help you find the range of the third side. The two given sides are 6 and 10, so the third side, \begin{align*}s\end{align*}, can either be the shortest side or the longest side. For example \begin{align*}s\end{align*} could be 5 because \begin{align*}6 + 5 > 10\end{align*}. It could also be 15 because \begin{align*}6 +10 > 15\end{align*}. Therefore, the range of values for \begin{align*}s\end{align*} is \begin{align*}4 < s < 16\end{align*}.

Notice the range is no less than 4, and not equal to 4. The third side could be 4.1 because \begin{align*}4.1 + 6 > 10\end{align*}. For the same reason, \begin{align*}s\end{align*} cannot be greater than 16, but it could 15.9, \begin{align*}10 + 6 > 15.9\end{align*}.

If two sides are lengths \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, then the third side, \begin{align*}s\end{align*}, has the range \begin{align*}a-b < s < a + b\end{align*}.

The SAS Inequality Theorem

The SAS Theorem compares two triangles. If we have two congruent triangles \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*}, marked below:

Therefore, if \begin{align*}AB = DE\end{align*} and \begin{align*}BC = EF\end{align*} and \begin{align*}m \angle B > m \angle E\end{align*}, then \begin{align*}AC > DF\end{align*}.

Now, let’s make \begin{align*}m \angle B > m \angle E\end{align*}. Would that make \begin{align*}AC > DF\end{align*}? Yes.

The SAS Inequality Theorem: If two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.

If \begin{align*}\overline{AB} \cong \overline{DE}, \overline{BC} \cong \overline{EF}\end{align*} and \begin{align*}m \angle B > m \angle E\end{align*}, then \begin{align*}\overline{AC} > \overline{DF}\end{align*}.

Example 5: List the sides in order, from least to greatest.

Solution: Let’s start with \begin{align*}\triangle DCE\end{align*}. The missing angle is \begin{align*}55^\circ\end{align*}. By Theorem 5-9, the sides, in order are \begin{align*}CE, \ CD\end{align*}, and \begin{align*}DE\end{align*}.

For \begin{align*}\triangle BCD\end{align*}, the missing angle is \begin{align*}43^\circ\end{align*}. Again, by Theorem 5-9, the order of the sides is \begin{align*}BD, \ CD\end{align*}, and \begin{align*}BC\end{align*}.

By the SAS Inequality Theorem, we know that \begin{align*}BC > DE\end{align*}, so the order of all the sides would be: \begin{align*}BD = CE, \ CD, \ DE, \ BC\end{align*}.

SSS Inequality Theorem

SSS Inequality Theorem: If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is greater in measure than the included angle of the second triangle.

If \begin{align*}\overline{AB} \cong \overline{DE}, \ \overline{BC} \cong \overline{EF}\end{align*} and \begin{align*}\overline{AC} > \overline{DF}\end{align*}, then \begin{align*}m \angle B > m \angle E\end{align*}.

Example 6: If \begin{align*}\overline{XM}\end{align*} is a median of \begin{align*}\triangle XYZ\end{align*} and \begin{align*}XY > XZ\end{align*}, what can we say about \begin{align*}m \angle 1\end{align*} and \begin{align*}m \angle 2\end{align*}?

Solution: \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{YZ}\end{align*}, so \begin{align*}YM = MZ\end{align*}. \begin{align*}MX = MX\end{align*} by the Reflexive Property and we know \begin{align*}XY > XZ\end{align*}.

We can use the SSS Inequality Theorem to say \begin{align*}m \angle 1 > m \angle 2\end{align*}.

Example 7: List the sides of the two triangles in order, from least to greatest.

Solution: There are no congruent sides or angles. Look at each triangle separately.

\begin{align*}\triangle XYZ\end{align*}: The missing angle is \begin{align*}42^\circ\end{align*}. By Theorem 5-9, the order of the sides is \begin{align*}YZ, \ XY\end{align*}, and \begin{align*}XZ\end{align*}.

\begin{align*}\triangle WXZ\end{align*}: The missing angle is \begin{align*}55^\circ\end{align*}. The order is \begin{align*}XZ, \ WZ\end{align*}, and \begin{align*}WX\end{align*}.

Because the longest side in \begin{align*}\triangle XYZ\end{align*} is the shortest side in \begin{align*}\triangle WXZ\end{align*}, we can put all the sides together in one list: \begin{align*}YZ, \ XY, \ XZ, \ WZ, \ WX\end{align*}.

Example 8: Below is isosceles triangle \begin{align*}\triangle ABC\end{align*}. List everything you can about the triangle and why.

Solution:

\begin{align*}AB = BC\end{align*} because it is given.

\begin{align*}m \angle A = m \angle C\end{align*} by the Base Angle Theorem.

\begin{align*}AD < DC\end{align*} because \begin{align*}m \angle ABD < m \angle CBD\end{align*} and the SAS Triangle Inequality Theorem.

Know What? Revisited The blue plane is further away from LAX because \begin{align*}110^\circ < 130^\circ\end{align*}. (SAS Inequality Theorem)

Review Questions

  • Questions 1-9 are similar to Examples 1 and 2.
  • Questions 10-18 are similar to Example 3.
  • Questions 19-26 are similar to Example 4.
  • Questions 27 and 28 are similar to Examples 5 and 6.
  • Question 29 is similar to Example 7.
  • Question 30 is similar to Example 8.

For questions 1-3, list the sides in order from shortest to longest.

For questions 4-6, list the angles from largest to smallest.

  1. Draw a triangle with sides 3 cm, 4 cm, and 5 cm. The angle measures are \begin{align*}90^\circ, \ 53^\circ\end{align*}, and \begin{align*}37^\circ\end{align*}. Place the angle measures in the appropriate spots.
  2. Draw a triangle with angle measures \begin{align*}56^\circ, \ 54^\circ\end{align*} and the included side is 8 cm. What is the longest side of this triangle?
  3. Draw a triangle with sides 6 cm, 7 cm, and 8 cm. The angle measures are \begin{align*}75.5^\circ, \ 58^\circ\end{align*}, and \begin{align*}46.5^\circ\end{align*}. Place the angle measures in the appropriate spots.

Determine if the sets of lengths below can make a triangle. If not, state why.

  1. 6, 6, 13
  2. 1, 2, 3
  3. 7, 8, 10
  4. 5, 4, 3
  5. 23, 56, 85
  6. 30, 40, 50
  7. 7, 8, 14
  8. 7, 8, 15
  9. 7, 8, 14.99

If two lengths of the sides of a triangle are given, determine the range of the length of the third side.

  1. 8 and 9
  2. 4 and 15
  3. 20 and 32
  4. 2 and 5
  5. 10 and 8
  6. \begin{align*}x\end{align*} and \begin{align*}2x\end{align*}
  7. The base of an isosceles triangle has length 24. What can you say about the length of each leg?
  8. The legs of an isosceles triangle have a length of 12 each. What can you say about the length of the base?
  9. What conclusions can you draw about \begin{align*}x\end{align*}?
  10. Compare \begin{align*}m \angle 1\end{align*} and \begin{align*}m \angle 2\end{align*}.
  11. List the sides from shortest to longest.
  12. Compare \begin{align*}m \angle 1\end{align*} and \begin{align*}m \angle 2\end{align*}. What can you say about \begin{align*}m \angle 3\end{align*} and \begin{align*}m \angle 4\end{align*}?

Review Queue Answers

  1. \begin{align*}4x-9 \le 19\!\\ {\;} \ \ \ 4x \le 28\!\\ {\;} \quad \ \ x \le 7\end{align*}
  2. \begin{align*}{\;} - 5>-2x+13\!\\ -18 > -2x\!\\ {\;} \ \ 9 < x\end{align*}
  3. \begin{align*}\frac{2}{3} x+1 \ge 13\!\\ {\;} \quad \ \ \frac{2}{3}x \ge 12\!\\ {\;} \qquad \ x \ge 18\end{align*}
  4. \begin{align*}-7<3x-1<14\!\\ -6<3x<15\!\\ -2<x<5\end{align*}

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