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Section 5-5
INDIRECT PROOF AND INEQUALITIES IN
ONE TRIANGLE
Indirect Proof – you begin by assuming that the
conclusion in false. Then, you show your assumption
is false.
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.
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.
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°
A triangle cannot have an angle with a measure of
zero degrees, therefore this conjecture is false.
Therefore, a triangle cannot have two right angles.
In other words, the longest side is opposite of
the largest angle. The shortest side is opposite of
the smallest angle. And the medium side is
opposite of the medium angle.
The converse is also true! The largest angle is
opposite the largest side, etc.
Example 2:
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.
Example 3:
Write the sides in order from
shortest to longest.
First…find the missing angle…
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
.
Write all of this down…
In other words, the two shortest sides of a triangle
must add up to be LARGER than the third side.
As shown below…. (you don’t have to write this down)
4+4=8, which is larger
than 7, so this can be
a triangle.
3-3=6, which is NOT
larger than 7, so this
CANNOT be a triangle.
Example 4:
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.
Example 5:
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.

Example 6: 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.
Now…watch this video for a little extra help…
Section 5-5 Video
Assignment #57
DUE TOMORROW!
Pg. 335
#2-14 even