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Name ________________________________________ Date __________________ Class__________________ LESSON 3-3 Reteach Proving Lines Parallel Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. You can use the Converse of the Corresponding Angles Postulate to show that two lines are parallel. Given: ∠1 ≅ ∠3 ∠1 ≅ ∠3 q || r ∠1 ≅ ∠3 are corresponding angles. Converse of the Corresponding Angles Postulate Given: m∠2 = 3x°, m∠4 = (x + 50)°, x = 25 m∠2 = 3(25)° = 75° Substitute 25 for x. m∠4 = (25 + 50)° = 75° Substitute 25 for x. m∠2 = m∠4 Transitive Property of Equality ∠2 ≅ ∠4 Definition of congruent angles q || r Converse of the Corresponding Angles Postulate For Exercises 1 and 2, use the Converse of the Corresponding Angles Postulate and the given information to show that c || d. 1. Given: ∠2 ≅ ∠4 2. Given: m∠1 = 2x°, m∠3 = (3x − 31)°, x = 31 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 3-22 Holt McDougal Geometry Name ________________________________________ Date __________________ Class__________________ LESSON 3-3 Reteach Proving Lines Parallel continued You can also prove that two lines are parallel by using the converse of any of the other theorems that you learned in Lesson 3-2. Theorem Hypothesis Converse of the Alternate Interior Angles Theorem Conclusion a || b ∠2 ≅ ∠3 Converse of the Alternate Exterior Angles Theorem f || g ∠1 ≅ ∠4 Converse of the Same-Side Interior Angles Theorem s || t m∠1 + m∠2 = 180° For Exercises 3–5, use the theorems and the given information to show that j || k. 3. Given: ∠4 ≅ ∠5 4. Given: m∠3 = 12x°, m∠5 = 18x°, x = 6 5. Given: m∠2 = 8x°, m∠7 = (7x + 9)°, x = 9 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 3-23 Holt McDougal Geometry supplementary by the Linear Pair Theorem. Therefore ∠3 and ∠2 must be congruent by the Congruent Supplements Theorem. Since ∠3 and ∠2 are congruent, HI and JK are parallel by the Converse of the Corresponding Angles Postulate. Practice C 1. x = 11; y = −5; m∠1 = 57°; m∠2 = 57°; m∠3 = 123° Reteach 1. ∠2 ≅ ∠4 ∠2 and ∠4 are corr. ∠s . c || d Conv. of Corr. ∠s Post. 2. m∠1 = 2x° = 2(31)° = 62° m∠3 = (3x − 31)° = 3(31)° − 31° = 62° m∠1 = m∠3 ∠1 ≅ ∠3 c || d Substitute 31 for x. Substitute 31 for x. Trans. Prop. of = Def. of ≅ ∠s Conv. of Corr. ∠s Post. 3. ∠4 ≅ ∠5 ∠4 and ∠5 are alt. int. ∠s . 2. j || k HJJG Possible answer: Construct FG through point C and parallel to AB . ∠3 and ∠4 are a linear pair, so m∠3 + m∠4 = 180° by the Linear Pair Theorem. But the Angle Addition Postulate shows that m∠4 = m∠ACF + m∠FCD, so by substitution m∠3 + m∠ACF + m∠FCD = 180°. m∠1 = m∠ACF by the Alternate Interior Angles Theorem and m∠2 = m∠FCD by the Corresponding Angles Postulate. Therefore m∠1 + m∠2 + m∠3 = 180° by substitution. 3. The measures of the segments are equal. Conv. of Alt. Int. ∠s Thm. 4. m∠3 = 12(6)° = 72° Substitute 6 for x. m∠5 = 18(6)° = 108° Substitute 6 for x. m∠3 + m∠5 = 72° + 108° = 180° Add angle measures. j || k Conv. of Same-Side Int. ∠s Thm. 5. m∠2 = 8(9)° = 72° m∠7 = 7(9)° + 9° = 72° m∠2 = m∠7 ∠2 ≅ ∠7 j || k Substitute 9 for x. Substitute 9 for x. Trans. Prop. of = Def. of ≅ ∠s Conv. of Alt. Ext. ∠s Thm. Challenge 1. a = 22.5 2. a = 13 3. a = 22 4. Explanations may vary. 5. a. Explanations may vary. b. 0 < c < 20; 0 < d < 100 4. Possible answer: If a triangle is isosceles, then the sides opposite the congruent angles are congruent. 6. a. m∠1 = m∠2 = m∠6 = (180 − p)° m∠3 = m∠4 = m∠5 = (180 − q)° m∠7 = p°; m∠8 = q° m∠9 = (180 − p–q) b. 0 < q < 90; p < q Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A24 Holt McDougal Geometry