Triangle Inequalities

1
Triangle Inequalities

• 4.6
• Apply inequalities in one triangle.

2
The positions of the longest and shortest sides
of a triangle are related to the positions of the
largest and smallest angles.
3
Example 1 Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from smallest to
largest.
The angles from smallest to largest are ?F, ?H
and ?G.
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Example 2 Ordering Triangle Side Lengths and
Angle Measures
Write the sides in order from shortest to longest.
m?R 180 (60 72) 48
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A triangle is formed by three segments, but not
every set of three segments can form a triangle.
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A certain relationship must exist among the
lengths of three segments in order for them to
form a triangle.
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Example 3 Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
8, 13, 21
7, 10, 19
Noby the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
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Example 4 Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
2.3, 3.1, 4.6
?
?
?
Yesthe sum of each pair of lengths is greater
than the third length.
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Example 5 Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
n 6, n2 1, 3n, when n 4.
Step 1 Evaluate each expression when n 4.
n 6
n2 1
3n
4 6
(4)2 1
3(4)
10
15
12
10
Example 5 Continued
Step 2 Compare the lengths.
?
?
?
Yesthe sum of each pair of lengths is greater
than the third length.
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Example 6 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 gt 13
x 13 gt 8
8 13 gt x
x gt 5
x gt 5
21 gt x
Combine the inequalities. So 5 lt x lt 21. The
length of the third side is greater than 5 inches
and less than 21 inches.
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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 gt 51
x 51 gt 46
46 51 gt x
? Inequal. Thm.
x gt 5
x gt 5
97 gt x
Subtr. Prop. of Inequal.
5 lt x lt 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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Check It Out! 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 gt 50
x 50 gt 22
22 50 gt x
? Inequal. Thm.
x gt 28
x gt 28
72 gt x
Subtr. Prop. of Inequal.
28 lt x lt 72
Combine the inequalities.
The distance from Seguin to Johnson City is
greater than 28 miles and less than 72 miles.
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Homework

• Pg 216 1-20
• Extra help and problems are on the next slides.
  Go to my website to see them.

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Lesson Quiz Part I
1. Write the angles in order from smallest to
largest. 2. Write the sides in order from
shortest to longest.
?C, ?B, ?A
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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. 4. Tell whether a triangle
can have sides with lengths 2.7, 3.5, and 9.8.
Explain.
5 cm lt x lt 29 cm
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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Check It Out! Example 1
Write the angles in order from smallest to
largest.
The angles from smallest to largest are ?B, ?A,
and ?C.
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Check It Out! Example 2
Write the sides in order from shortest to longest.
m?E 180 (90 22) 68
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Check It Out! Example 3
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 gt 17
x 17 gt 22
22 17 gt x
x gt 5
x gt 5
39 gt x
Combine the inequalities. So 5 lt x lt 39. The
length of the third side is greater than 5 inches
and less than 39 inches.
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Example

• A landscape architect is designing a triangular
  deck. She wants to place benches in the two
  largest corners. In which corners should she
  place the benches?

27ft
?
21ft
19ft
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Homework

• P. 216 1-24