133
Chapter 5
10. Given:
m
±
/
n
Prove:
Lines
m
and
n
intersect at exactly
one point.
Proof:
Case 1:
m
and
n
intersect at more than one point.
Step 1
Assume that
m
and
n
intersect at more
than one point.
Step 2
Lines
m
and
n
intersect at points
P
and
Q
.
Both lines
m
and
n
contain
P
and
Q
.
Step 3
By postulate, there is exactly one line
through any two points. Thus the
assumption is false, and lines
m
and
n
intersect in no more than one point.
Case 2:
m
and
n
do not intersect.
Step 1
Assume that
m
and
n
do not intersect.
Step 2
If lines
m
and
n
do not intersect, then
they are parallel.
Step 3
This conclusion contradicts the given
information. Therefore the assumption is
false, and lines
m
and
n
intersect in at
least one point. Combining the two cases,
lines
m
and
n
intersect in no more than
one point and no less than one point. So
lines
m
and
n
intersect in exactly one
point.
11. Given:
n
ABC
is a right triangle;
/
C
is a right
angle.
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 Fall '10
 Dr.Zhan
 Calculus, PreCalculus, Right triangle, Hypotenuse, triangle, AB AC

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