Parallel lines theorem
- 1. Masalah Kesejajaran Garis Zuliyana Dewi A. Mifta Zuliyanti Asti Pujiningtiyas Eva Wulansari
- 2. The Parallel Line Postulat • 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 p q 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 1. Suppose p ll r 2. There is a point common to p and r. Call it A. 3. p ll q 4. p 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 1. Indirect proof assumption 2. Restatement of 1 3. Given 4. Given 5. Statement 3 and 4 6. Logic of indirect proof
- 6. More Theorems About Parallel Lines p q 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 angels are congruent
- 8. 2 1 • Given : Lines p ll q with transversal r. 1 and 2 are alternate interior angles. • Prove : • Plan : Assume that 1 is not congruent to 2 and find a contradiction. p q A B r 2 1 p q A B r
- 9. Statements Reasons 1. is not congruent to 1. Indirect proof assumption 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.
- 10. • If two parallel lines are cut by a transversal, then alternate exterior angles are congruent • If two parallel lines are cut by a transversal, then corresponding angles are congruent • If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary