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Masalah Kesejajaran Garis Kelompok • Asti Pujiningtyas • Eva Wulansari • Mifta Zuliyanti • Zuliyana Dewi A. 6 4101414009 4101414023 4101414016 4101414001 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