Warm Up(Add to Notes)

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Warm Up(Add to Notes) 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
2
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Indirect Proof and Inequalities in One Triangle
Holt Geometry
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In an indirect proof, you assume 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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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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Write the angles in order from smallest to
largest.
The angles from smallest to largest are ?F, ?H
and ?G.
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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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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 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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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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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.