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Students will be able to ◦ Determine whether two lines are parallel ◦ Write flow proofs ◦ Define and apply the converse of the theorems from the previous section You can use certain angle pairs to determine if two lines are parallel What is the corresponding angles theorem? If a transversal intersects two parallel lines, then corresponding angles are congruent What is the converse of the corresponding angles theorem? If two lines and a transversal form congruent corresponding angles, then the lines are parallel Which lines are parallel if <6 ≅ <7? ◦ m || l Which lines are parallel if <4 ≅ <6 ◦ a || b If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel If two lines and a transversal form same side interior angles that are supplementary, then the two lines are parallel. If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. If corresponding angles are congruent, then the lines are parallel If alternate interior lines are congruent, then the lines are parallel If alternate exterior lines are congruent, then the lines are parallel If same side interior angles are supplementary, then the lines are parallel In order to use the theorems relating to parallel lines, you must first prove the lines are parallel if it is not given/stated in the problem. Even if lines appear to be parallel, you cannot assume they are parallel Always assume diagrams are NOT drawn to scale, unless otherwise stated Third way to write a proof In a flow proof, arrows show the flow, or the logical connections, between statements. Reasons are written below the statements Given: <4 ≅ <6 Prove: l || m <4 ≅ <6 Given <2 ≅ <6 <2 ≅ <4 Vert. <s are ≅ Trans. Prop of ≅ L || m Converse of Corresponding Angles Thm. *You cannot use the Corresponding Angles Thm to say <2 ≅ <6 because we do not know if the lines are parallel Given: m<5 = 40, m<2 = 140 Prove: a || b Start with what you know ◦ The given statement ◦ What you can conclude from your picture. What you need to know ◦ Which theorem you can use to show a||b Given: m<5 = 40, m<2 = 140 Prove: a || b <5 = 40 Given <5 and <2 are Supp. <s Def. of Supp. <s <2 = 140 Given <5 and <2 are Same side Interior Angles Def. of Same Side Interior <s a || b Converse of Same Side Int. <s Thm You now have four ways to prove if two lines are parallel What is the value of x for which a || b? Work backwards. What must be true of the given angles for a and b to be parallel? How are the angles related? ◦ Same side interior ◦ Therefore, they must add to be 180 What is the value of x for which a || b? Work backwards. What must be true of the given angles for a and b to be parallel? How are the angles related? ◦ Corresponding Angles ◦ Therefore, the angles are congruent Pg. 160 – 162 # 7 – 16, 21 – 24, 28, 32 16 Problems