Name
LESSON
5-5
Date
Class
Reteach
Indirect Proof and Inequalities in One Triangle
In a direct proof, you begin with a true hypothesis and prove that a conclusion is true. In an
indirect proof, you begin by assuming that the conclusion is false (that is, that the opposite
of the conclusion is true). You then show that this assumption leads to a contradiction.
Consider the statement “Two acute angles do not form a linear pair.”
Writing an Indirect Proof
Steps
Example
1. Identify the conjecture to be proven.
Given: 1 and 2 are acute angles.
Prove: 1 and 2 do not form a linear pair.
2. Assume the opposite of the conclusion
is true.
Assume 1 and 2 form a linear pair.
3. Use direct reasoning to show that the
assumption leads to a contradiction.
m1 m2 180° by def. of linear pair.
Since m1 90° and m2 90°,
m1 m2 180°.
This is a contradiction.
The assumption that 1 and 2 form a
linear pair is false. Therefore 1 and 2 do
not form a linear pair.
4. Conclude that the assumption is false
and hence that the original conjecture
must be true.
Use the following statement for Exercises 1–4.
#
An obtuse triangle cannot have a right angle.
1. Identify the conjecture to be proven.
"
!
Given: ABC is an obtuse , B is an obtuse angle; Prove: ABC
does not have a right angle.
2. Assume the opposite of the conclusion. Write this assumption.
Assume ABC does have a right angle. Let A be a right angle.
3. Use direct reasoning to arrive at a contradiction.
Possible answer: If A is a right angle, then mB mC 90°.
But mB > 90°, since B is obtuse. So this is a contradiction.
4. What can you conclude?
The assumption that ABC does have a right angle is false. Therefore
ABC does not have a right angle.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
38
Holt Geometry
Name
LESSON
5-5
Date
Class
Reteach
Indirect Proof and Inequalities in One Triangle
Theorem
continued
Example
X is the
largest angle.
9
If two sides of a triangle are not
congruent, then the larger angle is
opposite the longer side.
_
WY is the
longest side.
7
If WY XY, then mX mW.
Another similar theorem says that if two angles of a triangle are not congruent,
then the longer side is opposite the larger angle.
Write the correct answer.
(
4
—
—
*
3
—
+
6
5. Write the angles in order from smallest
to largest.
6. Write the sides in order from shortest
to longest.
_ _ _
V, S, T
JK, KH, HJ
Theorem
Example
Triangle Inequality Theorem
The sum of any two side lengths of a
triangle is greater than the third side
length.
abc
B
bca
A
cab
C
Tell whether a triangle can have sides with the given lengths. Explain.
7. 3, 5, 8
8. 11, 15, 21
No; 3 5 8, which is not
Yes; the sum of each pair of
greater than the length of the
lengths is greater than the length
third side.
of the third side.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
39
Holt Geometry