Geom_3eTE.03F4.X_145-146 3/24/06 11:17 AM Page 145
GPS
Guided Problem Solving
Guided Problem Solving
Understanding
Proof Problems
FOR USE WITH PAGE 143, EXERCISE 3
Understanding Proof Problems Read the problem below. Then let Collin’s
thinking guide you through the solution. Check your understanding with the
exercise at the bottom of the page.
Completing a proof fully and
correctly requires connecting
steps logically and justifying each
step. Here, the process of proof is
analyzed using Theorem 3-9.
Copy and complete the paragraph proof of Theorem 3-9 for three coplanar lines.
If two lines are parallel to the same line, then they are parallel to each other.
1
Given: / 6 k and m 6 k
Prove: / 6 m
k
Proof: / 6 k means that &2 > &1 by the a. 9 Postulate.
m 6 k means that b. 9 > c. 9 for the same reason. By the
Transitive Property of Congruence, &2 > &3. By the d. 9 Postulate, / 6 m.
ᐉ
2
Teaching Notes
3
Proofs may be new to your
students. A good introduction is
to analyze a proof with missing
parts and to recreate the reasoning
behind each step. Comparing
“What Collin Thinks” and “What
Collin Writes” allows students to
see the logical development of
each answer.
m
What Collin Writes
What Collin Thinks
< n k means that l2 O l1 by the
a. Corresponding Angles Postulate.
Let’s see. &1 and &2 are corresponding angles.
If / 6 k, then &1 > &2 by Postulate 3-1 on
page 128.
Tactile Learners
Now I will use the fact that m 6 k. So I’ll
ignore line O. That leaves me with &1 and &3
to consider. These also are corresponding angles.
Now, &2 > &3 and they are corresponding
angles. This fits the postulate in which the
hypothesis is “corresponding angles are
congruent.” By Postulate 3-2 on page 134,
I can conclude “the lines are parallel.”
Have students use three large
rulers on a desk to model lines
k, l, and m.
m n k means that b. l1 O c. l3 for the
same reason.
Teaching Tip
Point out that all of the information given in the theorem is used
in the proof. Explain that this is
normally the case when proving
a theorem.
By the Transitive Property of Congruence,
l2 O l3. By the
d. Converse of the Corresponding Angles
Postulate, < n m.
Error Prevention!
Students may confuse the
Corresponding Angles Postulate
and its converse. Ask them to
identify the hypotheses and
conclusions of each.
EXERCISE
Copy and complete this paragraph proof of Theorem 3-9 for three coplanar lines.
If two lines are parallel to the same line, then they are parallel to each other.
p
1
Given: q 6 p and p 6 r
Exercise
Have students work independently
to complete the proof. 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 complete a proof.
2
Prove: q 6 r
Proof: q 6 p means that &1 > &2 by the a. 9 Theorem. a. Alt. Int. '
p 6 r means that b. 9 > c. 9 by the Corresponding Angles Postulate. b. l2 c. l3
By the Transitive Property of Congruence, &1 > &3. By the d. 9 Theorem, q 6 r.
d. Conv. of the Alt. Int. '
Guided Problem Solving
true; however, when
a n b, and b n a, it does
not follow that a n a.
23. Reflexive: a a; false; lines are two lines that
intersect to form right '.
r
Symmetric: If a b, then
b a; true; b and a
intersect to form right '.
Transitive: If a b, and
b c; then a c; false;
in a plane, two lines to
the same line are n.
q
3
Understanding Proof Problems
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