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Masalah Kesejajaran Garis
Kelompok
• Asti Pujiningtyas
• Eva Wulansari
• Mifta Zuliyanti
• Zuliyana Dewi A.
6
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The Parallel Line Postulate
• Given a line l and a point P not on l, there exist
only one line thought P parallel to l
•P
l
Construct a line through P that is parallel to l
• Given lines p, q and r
if p ll q and q ll r , then p ll r
p
q
r
Proof
• Given : Lines p, q, and r are three distinct
lines, p ll q, q ll r
• Prove : p ll r
p
q
r
A
Statement
1. Suppose p ll r
2. There is a point common to
p and r. Call it A.
3. p ll q
4. r ll q
5. Lines p and r are two distinct
lines through A parallel to q
(contradiction of the Parallel
Postulate which assert that
there is only one line
thought A parallel to q)
6. Therefore p ll r
Reason
1. Indirect proof assumption
2. Restatement of 1
3. Given
4. Given
5. Statement 3 and 4
6. Logic of indirect proof
More Theorems About
Parallel Lines
1
2
p
q
1
2
p
q
p
Given : p ǁ q
Observe that
1
2
q
• If two parallel lines are cut by a transversal,
then alternate interior angles are congruent
r
A
2
1
B
p
q
• Given : Lines p ll q with transversal r.
and
are alternate interior angles.
• Prove :
• Plan : Assume that
is not congruent to
and find a contradiction.
r
A
2
B
1
p
q
Statements
Reasons
1.
1. Indirect proof assumption
is not congruent to
2. Construct a line through A so that
, and
and
are
alternate interior angels.
2. Construction
3.
, A is on s.
3. If alternate interior angels are
congruent, then the lines are parallel.
4.
, A is on p.
4. Given
5. There are two lines through A parallel
to q. (Contradiction of the Parallel
Postulate).
5. Statements 3 and 4
6.
6. Logic of indirect proof.
• If two parallel lines are cut by a transversal,
then alternate exterior angles are congruent
r
2
A
B
1
p
q
Given : Lines p ll q with transversal r.
and
are alternate exterior angles.
Prove :
Plan : Assume that
is not congruent to
and
find a contradiction.
• If two parallel lines are cut by a transversal,
then corresponding angles are congruent
r
2
A
1
B
p
q
Given : Lines p ll q with transversal r.
and
are corresponding angles.
Prove :
Plan : Assume that
is not congruent to
find a contradiction.
and
• If two parallel lines are cut by a transversal,
then the interior angles on the same side of the
transversal are supplementary
r
2
1
B
A
p
q
Given : Lines p ll q with transversal r.
and
are supplementary.
Prove :
is not congruent to
Plan : Assume that
is not congruent to
find a tautology.
and