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1 Lesson Plan #29 Class: Geometry HW 29: Date: Monday November 14th, 2011 Topic: Proving lines are parallel? Aim: How do we prove lines are parallel? Objectives: 1) Students will be able to prove lines parallel. Note: From Lesson 27 recall the theorem - The measure of an exterior angle of a triangle is greater than the measure of either non adjacent interior angle. Do Now: Examine this proof by contradiction or indirect proof. Sample Test Question: 1) Which type of proof begins by assuming the opposite of what you want to prove? A) Two column B) Indirect C) Paragraph D) Flow 2) If two lines are each parallel to a third line, then A) They are parallel to each other. B) They are perpendicular to each other. C) They are a pair of skew lines. D) Their relationship can’t be determined. PROCEDURE: Write the Aim and Do Now Get students working! Take attendance Give Back HW Collect HW Go over the Do Now Assignment #1: Go to http://www.mathopenref.com/transversal.html Drag one of the two lines cut by the transversal. What is true of the two lines cut by the transversal when the alternate interior angles are congruent? 2 Let’s try to prove that “if two coplanar lines are cut by a transversal so that alternate interior angles formed are congruent, then the two lines are parallel.” Given: ⃡𝐴𝐵 and ⃡𝐶𝐷 are cut by transversal ⃡𝐸𝐹 at points E and F respectively. <1 ≅< 2 Prove: ⃡𝐴𝐵 ∥ ⃡𝐶𝐷 4. Statements ⃡𝐴𝐵 and ⃡𝐶𝐷 are cut by transversal ⃡𝐸𝐹 at points E and F respectively Let ⃡𝐴𝐵 not be parallel to ⃡𝐶𝐷 ⃡𝐴𝐵 and ⃡𝐶𝐷 intersect at some point P, forming Δ𝐸𝐹𝑃 𝑚<1 >𝑚<2 5. 6. 7. But < 1 ≅ < 2 𝑚<1 ≅ 𝑚<2 ⃡𝐴𝐵 ∥ ⃡𝐶𝐷 1. 2. 3. Reasons 1. Given 2. 3. Assumption If coplanar lines are not parallel, then they are intersecting. 4. The measure of an exterior angle of a triangle is greater than the measure of either non adjacent interior angle 5.Given 6. Congruent angles are equal in measure 7. Contradiction in steps 4 and 6, therefore the assumption in step 2 is false. Theorem: If two coplanar lines are cut by a transversal, so that alternate interior angles formed are congruent, then the two lines are parallel. Let’s examine the two interior angles on the same side of the transversal at http://www.mathopenref.com/transversal.html What can you tell about those two angles? Let’s see if we can prove that if two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, then the lines are parallel. Finish the proof Given: ⃡ intersects ⃡𝐴𝐵 and ⃡𝐶𝐷 . 𝐸𝐹 < 5 is the supplement of < 4 Prove: ⃡𝐴𝐵 ∥ ⃡𝐶𝐷 Statements ⃡ 1.𝐸𝐹 intersects ⃡𝐴𝐵 and ⃡𝐶𝐷 . 2.< 3 and < 4 form a linear pair 3. 𝑚 < 3 + 𝑚 < 4 = 180 4.< 3 is the supplement of < 4 5.< 5 is the supplement of < 4 6. < 3 ≅< 5 ⃡ ∥ ⃡𝐶𝐷 7.𝐴𝐵 Reasons 1.Given 2. A linear pair of angles are two adjacent angles whose sum is a straight angle (1) 3. A straight angle is an angle whose degree measure is 180o. (1) 4. Supplementary angles are two angles the sum of whose degree measures is 180o. (2) 5. Given 6. 7. Theorem: If two coplanar lines are cut by a transversal so that the interior angles on the same side of the transversal are supplementary, then the lines are parallel. 3 Other ways to prove lines are parallel (presented without proof) Theorem: If two coplanar lines are cut by a transversal, so that corresponding angles are congruent, then the two lines are parallel Theorem: If two lines are perpendicular to the same line, then they are parallel. Summary of ways to prove lines parallel 1) A pair of alternate interior angles congruent 2) A pair of corresponding angles congruent 3) A pair of interior angles on the same side of the transversal are supplementary 4) Both lines are perpendicular to the same line 5) Both lines are parallel to the same line. 6) If two lines are cut by a transversal forming a pair of alternate exterior angles congruent, then the two lines are parallel Assignment: 4