5-5 indirect proof and inequalities in one triangle
TRANSCRIPT
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle 5-5 Indirect Proof and Inequalities
in One Triangle
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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 > 6, then 2x >
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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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
Apply inequalities in one triangle.
Objectives
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Holt McDougal Geometry
5-5 Indirect Proof and 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.
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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.
The shortest side is , so the smallest angle is F.
The longest side is , so the largest angle is G.
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Holt McDougal Geometry
5-5 Indirect Proof and 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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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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.
The shortest side is , so the smallest angle is B.
The longest side is , so the largest angle is C.
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
Check It Out! 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
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
A triangle is formed by three segments, but not every set of three segments can form a triangle.
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Holt McDougal Geometry
5-5 Indirect Proof and 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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Holt McDougal Geometry
5-5 Indirect Proof and 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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Holt McDougal Geometry
5-5 Indirect Proof and 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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Holt McDougal Geometry
5-5 Indirect Proof and 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, n2 – 1, 3n, when n = 4.
Step 1 Evaluate each expression when n = 4.
n + 6
4 + 6
10
n2 – 1
(4)2 – 1
15
3n
3(4)
12
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Holt McDougal Geometry
5-5 Indirect Proof and 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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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
Check It Out! 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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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
Check It Out! 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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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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
4 – 2
2
t2 + 1
(4)2 + 1
17
4t
4(4)
16
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
Check It Out! 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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Holt McDougal Geometry
5-5 Indirect Proof and 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.
Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.
x + 8 > 13
x > 5
x + 13 > 8
x > –5
8 + 13 > x
21 > x
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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.
Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches.
x + 22 > 17
x > –5
x + 17 > 22
x > 5
22 + 17 > x
39 > x
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Holt McDougal Geometry
5-5 Indirect Proof and Inequalities
in One Triangle
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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Holt McDougal Geometry
5-5 Indirect Proof and 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.
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 cm < x < 29 cm