Download Indirect Proofs and Triangle Inequalities

Indirect
Proof
Inequalities
Indirect
Proofand
and
Inequalities
5-5
5-5 ininOne
Triangle
One
Triangle
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
GeometryGeometry
Holt
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.”
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
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Objectives
Write indirect proofs.
Apply inequalities in one triangle.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Vocabulary
indirect proof
Holt McDougal 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 McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt McDougal 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 McDougal 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 McDougal 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.
Holt McDougal Geometry
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
Holt McDougal Geometry
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 McDougal 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 McDougal 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 McDougal 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 McDougal 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.
The shortest side is
smallest angle is F.
The longest side is
, so the
, so the largest angle is G.
The angles from smallest to largest are F, H and G.
Holt McDougal 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.
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
Holt McDougal Geometry
.
Indirect Proof and Inequalities
5-5 in One Triangle
Check It Out! Example 2a
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is B.
The longest side is
, so the
, so the largest angle is C.
The angles from smallest to largest are B, A, and C.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 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
Holt McDougal 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 McDougal 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 McDougal 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 McDougal 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


Yes—the sum of each pair of lengths is greater
than the third length.
Holt McDougal Geometry

Indirect Proof and Inequalities
5-5 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
n2 – 1
3n
4+6
(4)2 – 1
3(4)
10
15
12
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 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.
Holt McDougal 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
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
Holt McDougal Geometry
Indirect Proof and Inequalities
5-5 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.
Holt McDougal Geometry

Indirect Proof and Inequalities
5-5 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
Holt McDougal Geometry
4t
4(4)
16
t2 + 1
(4)2 + 1
17
Indirect Proof and Inequalities
5-5 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.
Holt McDougal 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 McDougal 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.
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.
Holt McDougal 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 McDougal Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
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 > 50
x + 50 > 22
x > 28
28 < x < 72
x > –28
22 + 50 > x
Δ Inequal. Thm.
72 > x
Subtr. Prop. of
Inequal.
Combine the inequalities.
The distance from Seguin to Johnson City is greater
than 28 miles and less than 72 miles.
Holt McDougal 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 McDougal 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 McDougal Geometry