1. Masalah Kesejajaran Garis
Kelompok 6
Asti Pujiningtyas
Eva Wulansari
Mifta Zuliyanti
Zuliyana Dewi A

2. 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

3. Given lines p, q and r if p ll q and q ll r , then p ll r

4. Proof
Given : Lines p, q, and r are three distinct lines, p ll q, q ll r
Prove : p ll r A p q r

5. Statement
Reason
Suppose p ll r
There is a point common to p and r. Call it A.
p ll q
r ll q
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)
Therefore p ll r
Indirect proof assumption
Restatement of 1
Given
Statement 3 and 4
Logic of indirect proof

6. More Theorems About Parallel Linesq 1 2 2 1 p q 1 2 p q
Given : p ǁ q
Observe that

7. If two parallel lines are cut by a transversal, then alternate interior angles are congruent2 1 p q A B r
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.

8. 2 1 p q A B r
Statements
Reasons
is not congruent to
1. Indirect proof assumption
2. Construct a line through A so that
, and and are alternate interior angels.
2. Construction
, A is on s.
3. If alternate interior angels are congruent, then the lines are parallel.
, 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.

9. If two parallel lines are cut by a transversal, then alternate exterior angles are congruent2 1 p q A B r
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.

10. If two parallel lines are cut by a transversal, then corresponding angles are congruent2 1 p q A B r
Given : Lines p ll q with transversal r.
and are corresponding angles.
Prove :
Plan : Assume that is not congruent to and find a contradiction.

11. If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary 2 1 p q A B r
Given : Lines p ll q with transversal r.
and are supplementary.
Prove : is not congruent to
Plan : Assume that is not congruent to and find a tautology.