Download G5-5-Indirect Proof

Indirect
Proof
and
Inequalities
Indirect
Proof
and
Inequalities
5-5
5-5 in One Triangle
in One Triangle
Holt
Geometry
Holt
Geometry
Indirect Proof and Inequalities
5-5 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Objectives
Write indirect proofs.
Apply inequalities in one triangle.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Vocabulary
indirect proof
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Helpful Hint
When writing an indirect proof, look for a
contradiction of one of the following: the given
information, a definition, a postulate, or a
theorem.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 1: Writing an Indirect Proof
Write an indirect proof that if a > 0, then
Step 1 Identify the conjecture to be proven.
Given: a > 0
Prove:
Step 2 Assume the opposite of the conclusion.
Assume
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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
However, 1 > 0.
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Simplify.
Indirect Proof and Inequalities
5-5 in One Triangle
Example 1 Continued
Step 4 Conclude that the original conjecture is true.
The assumption that
Therefore
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is false.
Indirect Proof and Inequalities
5-5 in One Triangle
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 triangle’s 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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°.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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.
Holt Geometry
Indirect Proof and Inequalities
5-5 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
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2A: Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from
smallest to largest.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2B: Ordering Triangle Side Lengths and
Angle Measures
Write the sides in order from
shortest to longest.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
Holt Geometry
Indirect Proof and Inequalities
5-5 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 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
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 3a
Tell whether a triangle can have sides with the
given lengths. Explain.
8, 13, 21
Holt Geometry
Indirect Proof and Inequalities
5-5 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 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.
Holt Geometry
Indirect Proof and Inequalities
5-5 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
Indirect Proof and Inequalities
5-5 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.
Holt Geometry