Indirect Proof and Inequalities

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Indirect Proof and Inequalities in One Triangle
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up 1. Write a conditional from the
sentence An isosceles triangle has two congruent
sides. 2. Write the contrapositive of the
conditional If it is Tuesday, then John has a
piano lesson. 3. Show that the conjecture If
x gt 6, then 2x gt 14 is false by finding a
counterexample.
If a ? is isosc., then it has 2 ? sides.
If John does not have a piano lesson, then it is
not Tuesday.
x 7
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Objectives
Write indirect proofs. Apply inequalities in one
triangle.
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Vocabulary
indirect proof
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So far you have written proofs using direct
reasoning. You began with a true hypothesis and
built a logical argument to show that a
conclusion was true. In an indirect proof, you
begin by assuming that the conclusion is false.
Then you show that this assumption leads to a
contradiction. This type of proof is also called
a proof by contradiction.
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Example 1 Writing an Indirect Proof
Step 1 Identify the conjecture to be proven.
Given a gt 0
Step 2 Assume the opposite of the conclusion.
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Example 1 Continued
Step 3 Use direct reasoning to lead to a
contradiction.
Given, opposite of conclusion
Zero Prop. of Mult. Prop. of Inequality
1 ? 0
Simplify.
However, 1 gt 0.
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Example 1 Continued
Step 4 Conclude that the original conjecture is
true.
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Check It Out! Example 1
Write an indirect proof that a triangle cannot
have two right angles.
Step 1 Identify the conjecture to be proven.
Given A triangles interior angles add up to
180.
Prove A triangle cannot have two right angles.
Step 2 Assume the opposite of the conclusion.
An angle has two right angles.
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Check It Out! Example 1 Continued
Step 3 Use direct reasoning to lead to a
contradiction.
m?1 m?2 m?3 180
90 90 m?3 180
180 m?3 180
m?3 0
However, by the Protractor Postulate, a triangle
cannot have an angle with a measure of 0.
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Check It Out! Example 1 Continued
Step 4 Conclude that the original conjecture is
true.
The assumption that a triangle can have two right
angles is false.
Therefore a triangle cannot have two right angles.
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The positions of the longest and shortest sides
of a triangle are related to the positions of the
largest and smallest angles.
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Example 2A 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 2B 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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Check It Out! Example 2a
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 2b
Write the sides in order from shortest to longest.
m?E 180 (90 22) 68
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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 3A Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
7, 10, 19
Noby the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
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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
?
?
?
Yesthe sum of each pair of lengths is greater
than the third length.
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Example 3C 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
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Example 3C Continued
Step 2 Compare the lengths.
?
?
?
Yesthe sum of each pair of lengths is greater
than the third length.
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Check It Out! Example 3a
Tell whether a triangle can have sides with the
given lengths. Explain.
8, 13, 21
Noby the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
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Check It Out! Example 3b
Tell whether a triangle can have sides with the
given lengths. Explain.
6.2, 7, 9
?
?
?
Yesthe sum of each pair of lengths is greater
than the third side.
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Check It Out! Example 3c
Tell whether a triangle can have sides with the
given lengths. Explain.
t 2, 4t, t2 1, when t 4
Step 1 Evaluate each expression when t 4.
t 2
t2 1
4t
4 2
(4)2 1
4(4)
2
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16
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Check It Out! Example 3c Continued
Step 2 Compare the lengths.
?
?
?
Yesthe sum of each pair of lengths is greater
than the third length.
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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 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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Check It Out! 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 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 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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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.