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Geometry 3.3 Proving Lines are Parallel Goals Use postulates and theorems to prove two lines parallel. Solve problems with parallel lines. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 2 What we’ve been doing: Given two parallel lines cut by a transversal… July 7, 2017 Geometry 3.3 Proving Lines are Parallel 3 Corresponding Angles Congruent July 7, 2017 Geometry 3.3 Proving Lines are Parallel 4 Alternate Exterior Angles Congruent July 7, 2017 Geometry 3.3 Proving Lines are Parallel 5 Alternate Interior Angles Congruent July 7, 2017 Geometry 3.3 Proving Lines are Parallel 6 Same Side Interior Angles Supplementary July 7, 2017 Geometry 3.3 Proving Lines are Parallel 7 July 7, 2017 Geometry 3.3 Proving Lines are Parallel 8 Theorem: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Converse: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 9 In other words… Corr. s 2 lines || (Theorem 3.5, Corr. Converse) July 7, 2017 Geometry 3.3 Proving Lines are Parallel 10 Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Converse: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 11 In other words… Alt Int s 2 lines || (Theorem 3.6, Alternate Interior Angles Converse) July 7, 2017 Geometry 3.3 Proving Lines are Parallel 12 Theorem: If two parallel lines are cut by a transversal, then alternate Exterior angles are congruent. Converse: If two lines are cut by a transversal and alternate Exterior angles are congruent, then the lines are parallel. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 13 In other words… Alt Ext s 2 lines || (Theorem 3.7, Alternate Exterior Angles Converse) July 7, 2017 Geometry 3.3 Proving Lines are Parallel 14 Theorem: If two parallel lines are cut by a transversal, then same side interior angles are supplementary. Converse: If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 15 In other words… y° x + y = 180° x° SS Int s supp 2 lines || (Theorem 3.8, Same Side Interior Angles Converse) July 7, 2017 Geometry 3.3 Proving Lines are Parallel 16 To show two lines parallel, show that one of these is true: Corresponding angles congruent. Alternate interior angles congruent. Alternate exterior angles congruent. Same side interior angles supplementary. You only need one pair for any one of these reasons. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 17 To show two lines parallel, show that one of these is true: Corr. s Alt. Int. s Alt. Ext. s SS Int. s supp. You only need one pair for any one of these reasons. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 18 Example 1 Given: m t, n t. t Prove: m || n m 1 Not drawn to scale 2 n Obviously Proof: Since m t, 1 is a right angle. Since n t, 2 is a right angle. All right angles are congruent, so 1 2. This means m || n since alt int s 2 lines ||. July 7, 2017 Geometry 3.3 Proving Lines are Parallel 19 Example 2 Given: 5 6; 6 4 Prove: AD || BC 4 B A 5 6 D C Proof: If 5 6 and 6 4, then 5 4. So AD || BC July 7, 2017 Transitive Prop because of alt int s . Geometry 3.3 Proving Lines are Parallel 20 Example 3 m n (2x + 1)° (3x – 5)° Find the value of x to make m || n. These are alt int angles. 2x + 1 = 3x – 5 6=x July 7, 2017 Geometry 3.3 Proving Lines are Parallel 21 Write a proof In the diagram, p || q and ∠1 is supplementary to ∠2. Prove r ||s using a proof. Statements Reasons ∠1 is supp to ∠2 m ∠1 + m ∠2 = 180 p || q m∠2=m∠3 m ∠1 + m ∠ 3 = 180 ∠1 and ∠ 3 are supp r || s July 7, 2017 Given Def of supp ∠’s Given Alt Int ∠’s Theorem Substitution Def of supp ∠’s Same Side Int ∠’s Converse Geometry 3.3 Proving Lines are Parallel 22 Theorem 3.9 Transitive Property of parallel lines If two lines are parallel to the same line, then they are parallel to each other. m July 7, 2017 n p If m || n and p || n, then m || p. Geometry 3.3 Proving Lines are Parallel 23 Slat 1 is parallel to slat 2. Slat 2 is parallel to slate 3. 1 2 3 Theorem 3.9 Why is slat 1 parallel to slat 3? If two lines are parallel to the same line, then they are parallel to each other. July 7, 2017 Geometry 3.5 Using Properties of Parallel Lines 24