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Guided Problem Solving
Guided Problem Solving
Understanding
Proof Problems
FOR USE WITH PAGE 284, EXERCISE 21
Understanding Proof Problems Read the problem below. Then let Resa’s
thinking guide you through the solution. Check your understanding with the
exercises at the bottom of the page.
The technique of indirect proof
is an invaluable tool in many
mathematics situations, both in
geometry and in more advanced
mathematics studies. This feature
helps students understand how
indirect proofs are developed,
reinforces the negation of “at
most one,” and helps students
begin generating their own
indirect proofs.
Fill in the blanks to prove the following statement.
In a given triangle, #LMN, there is at most one right angle.
Given: #LMN
Prove: #LMN has at most one right angle.
Proof: Assume that #LMN has more than one a. 9. That is, assume that both
&M and &N are b. 9. If &M and &N are both right angles, then m&M =
m&N = c. 9. By the Triangle Angle-Sum Theorem, m&L + m&M + m&N =
d. 9. Use substitution to find m&L + e. 9 + f. 9 = 180. When you solve
for m&L, you find that m&L = g. 9. This means that there is no #LMN,
which contradicts the information you are given. So the assumption that
#LMN has h. 9 must be false. Therefore, #LMN has i. 9.
Guided Instruction
What Resa Writes
What Resa Thinks
The essence of indirect proof is
assuming that the negation of a
statement’s conclusion leads to
the negation of its hypothesis.
This is equivalent to proving the
contrapositive of the original
statement. Because a statement
and its contrapositive are logically
equivalent, proving one true is
equivalent to proving both true.
Assume that kLMN has more than one
a. right angle.
To prove #LMN has at most one right angle,
I will assume that it does not have at most one.
This means that it has more than one.
Now I’ll copy the next line in the proof. Huh?
Where did &M and &N come from? Oh, I see.
“more than one” means “at least two,” so I have
to choose at least two angles and assume they
are right angles.
The measure of a right angle is 90.
That is, assume that both lM and lN
are b. right angles.
If lM and lN are both right angles,
then mlM ≠ mlN ≠ c. 90.
The Triangle Angle-Sum Theorem says that the
sum of the measures of the angles in a triangle
is 180. “Use substitution.” I can substitute 90 for
m&M and for m&N. Solving for m&L is easy.
mlL ± mlM ± mlN ≠ d. 180.
Substitute to find
mlL ± e. 90 ± f. 90 ≠ 180.
mlL ≠ g. 0.
But wait! How can m&L = 0?
Oh! I remember. In an indirect proof, I’m
looking for a contradiction. Well, I’ve got one!
The rest is easy.
So the assumption that kLMN has
h. more than one right angle must be
false. Therefore kLMN has i. at most one
right angle.
Visual Learners
Have a student provide a diagram
for the class to follow as the proof
unfolds.
Teaching Tip
Ask the class for examples of
conditional statements and their
contrapositives. Ask: How could
you write the contrapositive of
“If figure LMN is a triangle, then it
contains at most one right angle”?
Sample: If figure LMN contains
more than one right angle, then
it is not a triangle.
Exercises
EXERCISES
1. Given: #ABC is a right triangle.
2. Given: #ABC is an obtuse triangle.
Prove: #ABC is not an obtuse triangle.
1-2. See margin.
Prove: #ABC is not a right triangle.
Guided Problem Solving Understanding Proof Problems
1. Suppose lA is the given
rt. l, and assume that
kABC has obtuse lB.
Thus mlA ± mlB is
greater than 90 ± 90 ≠ 180
which contradicts the
kl–Sum Theorem.
2. Contrapositive of
Exercise 1.
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Have students work independently
to prove the problems, showing
the steps they used. Then have
volunteers share with the class
what they were thinking as they
wrote each step. Elicit the fact
that there are often different
ways to arrive at the solution of a
problem.
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