5-5 Inequalities in One Triangle

5-5 Inequalities in One Triangle
1. Write a conditional from the sentence “An isosceles triangle has two congruent sides.”
If a ∆ is isosc., then it has 2  sides.
2. Write the contrapositive of the conditional “If it is Tuesday, then John has a piano lesson.”
If John does not have a piano lesson, then it is not Tuesday.
3. Show that the conjecture “If x > 6, then 2x >
14” is false by finding a counterexample.
x = 7
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5-5 Inequalities in One Triangle
Holt Geometry

5-5 Inequalities in One Triangle
The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.
Holt Geometry

5-5 Inequalities in One Triangle
Example 2A: Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from smallest to largest.
The shortest side is , so the smallest angle is  F.
The longest side is , so the largest angle is  G.
The angles from smallest to largest are  F,  H and  G.
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5-5 Inequalities in One Triangle
Example 2B: Ordering Triangle Side Lengths and
Angle Measures
Write the sides in order from shortest to longest.
m  R = 180° – (60° + 72°) = 48°
The smallest angle is  R, so the shortest side is .
The largest angle is  Q, so the longest side is .
The sides from shortest to longest are
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5-5 Inequalities in One Triangle
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Example 2a
Write the angles in order from smallest to largest.
The shortest side is , so the smallest angle is  B.
The longest side is , so the largest angle is  C.
The angles from smallest to largest are  B,  A, and  C.
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5-5 Inequalities in One Triangle
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Example 2b
Write the sides in order from shortest to longest.
m  E = 180° – (90° + 22°) = 68°
The smallest angle is  D, so the shortest side is .
The largest angle is  F, so the longest side is .
The sides from shortest to longest are
Holt Geometry

5-5 Inequalities in One Triangle
A triangle is formed by three segments, but not every set of three segments can form a triangle.
Holt Geometry

5-5 Inequalities in One Triangle
A certain relationship must exist among the lengths of three segments in order for them to form a triangle.
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5-5 Inequalities in One Triangle
Example 3A: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.
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5-5 Inequalities in One Triangle
Example 3B: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
2.3, 3.1, 4.6
 
Yes—the sum of each pair of lengths is greater than the third length.

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5-5 Inequalities in One Triangle
Example 3C: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
n + 6, n 2 – 1, 3n, when n = 4.
Step 1 Evaluate each expression when n = 4.
n + 6
4 + 6
10 n 2 – 1
(4) 2 – 1
15
3n
3 (4)
12
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5-5 Inequalities in One Triangle
Example 3C Continued
Step 2 Compare the lengths.
 
Yes—the sum of each pair of lengths is greater than the third length.

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5-5 Inequalities in One Triangle
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Example 3a
Tell whether a triangle can have sides with the given lengths. Explain.
8, 13, 21
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.
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5-5 Inequalities in One Triangle
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Example 3b
Tell whether a triangle can have sides with the given lengths. Explain.
6.2, 7, 9
 
Yes—the sum of each pair of lengths is greater than the third side.

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5-5 Inequalities in One Triangle
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Example 3c
Tell whether a triangle can have sides with the given lengths. Explain.
t – 2, 4t, t 2 + 1, when t = 4
Step 1 Evaluate each expression when t = 4.
t – 2
4 – 2
2
4
4t
(4)
16 t 2
(4) 2
+ 1
+ 1
17
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5-5 Inequalities in One Triangle
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Example 3c Continued
Step 2 Compare the lengths.
  
Yes—the sum of each pair of lengths is greater than the third length.
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5-5 Inequalities in One Triangle
Example 4: Finding Side Lengths
The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 8 > 13
x > 5
x + 13 > 8
x > –5
8 + 13 > x
21 > x
Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.
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5-5 Inequalities in One Triangle
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Example 4
The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 22 > 17
x > –5
x + 17 > 22
x > 5
22 + 17 > x
39 > x
Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches.
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5-5 Inequalities in One Triangle
Example 5: Travel Application
The figure shows the approximate distances between cities in California.
What is the range of distances from San Francisco to Oakland?
Let x be the distance from San Francisco to Oakland.
x + 46 > 51 x + 51 > 46 46 + 51 > x
Δ Inequal. Thm.
x > 5 x > –5 97 > x
Subtr. Prop. of
Inequal.
5 < x < 97
Combine the inequalities.
The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.
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5-5 Inequalities in One Triangle
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Example 5
The distance from San Marcos to Johnson City is
50 miles, and the distance from Seguin to San
Marcos is 22 miles. What is the range of distances from Seguin to Johnson City?
Let x be the distance from Seguin to Johnson City.
x + 22 > 50
x > 28
x + 50 > 22
x > –28
22 + 50 > x
72 > x
28 < x < 72
Combine the inequalities.
Δ Inequal. Thm.
Subtr. Prop. of
Inequal.
The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.
Holt Geometry

5-5 Inequalities in One Triangle
Lesson Quiz: Part I
1. Write the angles in order from smallest to largest.
 C,  B,  A
2. Write the sides in order from shortest to longest.
Holt Geometry

5-5 Inequalities in One Triangle
Lesson Quiz: Part II
3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side.
5 cm < x < 29 cm
4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain.
No; 2.7 + 3.5 is not greater than 9.8.
5. Ray wants to place a chair so it is
10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is greater than the third length.
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