3.3 Proving Lines are Parallel Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel Goals Use postulates and theorems to prove two lines parallel. Solve problems with parallel lines. July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel What we’ve been doing: Given two parallel lines cut by a transversal… July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Corresponding Angles Congruent July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Alternate Exterior Angles Congruent July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Alternate Interior Angles Congruent July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Same Side Interior Angles Supplementary July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel Now we want to prove lines are parallel. July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel 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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel In other words… Corr. s   2 lines || (Theorem 3.5, Corr.  Converse) July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel 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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel In other words… Alt Int s   2 lines || (Theorem 3.6, Alternate Interior Angles Converse) July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel 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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel In other words… Alt Ext s   2 lines || (Theorem 3.7, Alternate Exterior Angles Converse) July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel 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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel In other words… y° x + y = 180° x° SS Int s supp  2 lines || (Theorem 3.8, Same Side Interior Angles Converse) July 2, 2018 Geometry 3.3 Proving Lines are Parallel

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 2, 2018 Geometry 3.3 Proving Lines are Parallel

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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel Example 1 Given: m  t, n  t. Prove: m || n m n t 1 Not drawn to scale Obviously 2 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 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel Example 2 Given: 5  6; 6  4 Prove: A B C D 4 6 5 Proof: If 5  6 and 6  4, then 5  4. Transitive Prop So because of alt int s . July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel Example 3 Find the value of x to make m || n. m n These are alt int angles. 2x + 1 = 3x – 5 6 = x (2x + 1)° (3x – 5)° July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.3 Proving Lines are Parallel 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 Given m ∠1 + m ∠2 = 180 Def of supp ∠’s p || q Given   Alt Int ∠’s Theorem m ∠ 2 = m ∠ 3   m ∠1 + m ∠ 3 = 180 Substitution ∠1 and ∠ 3 are supp Def of supp ∠’s r || s Same Side Int ∠’s Converse July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Theorem 3.9 Transitive Property of parallel lines If two lines are parallel to the same line, then they are parallel to each other. If m || n and p || n, then m || p. m n p July 2, 2018 Geometry 3.3 Proving Lines are Parallel

Geometry 3.5 Using Properties of Parallel Lines Slat 1 is parallel to slat 2. Slat 2 is parallel to slate 3. Why is slat 1 parallel to slat 3? 1 2 3 Theorem 3.9 If two lines are parallel to the same line, then they are parallel to each other. July 2, 2018 Geometry 3.5 Using Properties of Parallel Lines