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Name: ________________________ Class: ___________________ Date: __________ Proofs of Parallel and Perpendicular Lines Short Answer 1. Give the missing reasons in this proof of the Alternate Interior Angles Theorem. Given: l Ä n Prove: ∠4 ≅ ∠6 Statments Reasons 1. l Ä n 1. Given 2. ∠2 ≅ ∠6 a. ? 3. ∠4 ≅ ∠2 b. ? 4. ∠6 ≅ ∠4 c. ? 1 ID: A Name: ________________________ ID: A 2. State the missing reasons in this proof. Given: ∠1 ≅ ∠5 Prove: p Ä r Statements Reasons 1. ∠1 ≅ ∠5 Given 2. ∠4 ≅ ∠1 a.____ 3. ∠4 ≅ ∠5 b.____ 4. p Ä r c.____ 3. The 8 rowers in the racing boat stroke so that the angles formed by their oars with the side of the boat all stay equal. Explain why their oars on either side of the boat remain parallel. 4. Suppose you have four identical pieces of wood like those shown below. If m∠b = 40° can you construct a frame with opposite sides parallel? Explain. 5. Find the measure of each interior and exterior angle. The diagram is not to scale. 2 Name: ________________________ ID: A 6. The fireworks technician has two rocket launchers, each with a base and stand in the form of an L. A diagonal trough on which the technician places a rocket joins the ends of each L. One launcher has a 4-inch base and 10-inch stand. The other has a 6-inch base and a 15-inch stand. Explain why two rockets launched from the two devices could follow parallel paths. Essay 7. Write a paragraph proof of this theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Given: r ⊥ s, t ⊥ s Prove: r Ä t 8. Write a two-column proof. Given: ∠2 and ∠5 are supplementary. Prove: l Ä m 3 Name: ________________________ ID: A 9. Find the values of the variables. Show your work and explain your steps. The diagram is not to scale. Other 10. Given m∠1 = m∠2, what can you conclude about the lines l, m, and n? Explain. 11. Justify the statement algebraically. In a triangle, if the sum of the measures of two angles is equal to the measure of the third angle, then the triangle is a right triangle. 12. Line p contains points A(–1, 4) and B(3, –5). Line q is parallel to line p. Line r is perpendicular to line q. What is the slope of line r? Explain. 4 ID: A Proofs of Parallel and Perpendicular Lines Answer Section SHORT ANSWER 1. ANS: a. Corresponding angles. b. Vertical angles. c. Transitive Property. PTS: 1 DIF: L2 REF: 3-1 Properties of Parallel Lines OBJ: 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-1 Example 3 KEY: alternate interior angles | Alternate Interior Angles Theorem | proof | reasoning | two-column proof | multi-part question 2. ANS: a. Vertical angles. b. Transitive Property. c. Alternate Interior Angles Converse. PTS: 1 DIF: L2 REF: 3-2 Proving Lines Parallel OBJ: 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-2 Example 1 KEY: two-column proof | proof | reasoning | corresponding angles | multi-part question 3. ANS: The rowers keep corresponding angles congruent. PTS: OBJ: STA: TOP: 4. ANS: 1 DIF: L3 REF: 3-2 Proving Lines Parallel 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3 NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a 3-2 Example 1 KEY: transversal | word problem | reasoning | parallel lines No. Explanations may vary. Sample: Placing three pieces together forms same-side interior angles with measures of 80°. Since 80 + 80 ≠ 180, they are not supplementary, so the sides are not parallel. PTS: OBJ: NAT: STA: KEY: 1 DIF: L3 REF: 3-3 Parallel and Perpendicular Lines 3-3.1 Relating Parallel and Perpendicular Lines NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1 NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-3 Example 1 word problem | problem solving | parallel lines 1 ID: A 5. ANS: m∠1 = m∠2 = m∠3 = 90, m∠4 = 122, m∠5 = m∠6 = 58, m∠8 = 32, m∠7 = m∠9 = 148 PTS: 1 DIF: L3 REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem OBJ: 3-4.2 Using Exterior Angles of Triangles NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4b KEY: Triangle Angle-Sum Theorem | exterior angle 6. ANS: Pointed in the same direction, the two launchers have equal slopes, so the rockets would be set up to follow parallel paths. PTS: 1 DIF: L3 REF: 3-7 Slopes of Parallel and Perpendicular Lines OBJ: 3-7.1 Slope and Parallel Lines NAT: NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 C.1b | NJ 4.2.12 C.1d | NJ 4.2.12 C.1e | NJ 4.3.12 C.1a | NJ 4.3.12 C.2 KEY: slope | slopes of parallel lines | word problem | problem solving | writing in math ESSAY 7. ANS: [4] [3] [2] [1] PTS: OBJ: NAT: STA: KEY: By the definition of ⊥, r ⊥ s implies m∠2 = 90, and t ⊥ s implies m∠6 = 90. Line s is a transversal. ∠2 and ∠6 are corresponding angles. By the Converse of the Corresponding Angles Postulate, r || t. correct idea, some details inaccurate correct idea, not well organized correct idea, one or more significant steps omitted 1 DIF: L4 REF: 3-3 Parallel and Perpendicular Lines 3-3.1 Relating Parallel and Perpendicular Lines NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1 NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-3 Example 2 paragraph proof | proof | reasoning | extended response | rubric-based question | perpendicular lines 2 ID: A 8. ANS: [4] [3] [2] [1] PTS: OBJ: STA: KEY: angles 9. ANS: [4] [3] [2] [1] Statements 1. ∠2 and ∠5 are supplementary 2. ∠3 ≅ ∠2 3. ∠3 and ∠5 are supplementary 4. l Ä m Reasons 1. Given 2. Vertical angles 3. Substitution 4. Converse of Same-Side Interior Angles Theorem correct idea, some details inaccurate correct idea, some statements missing correct idea, several steps omitted 1 DIF: L4 REF: 3-2 Proving Lines Parallel 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3 NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a two-column proof | proof | extended response | rubric-based question | parallel lines | supplementary w + 31 + 90 = 180, so w = 59º. Since vertical angles are congruent, y = 59º. Since supplementary angles have measures with sum 180, x = v = 121º. z + 68 + y = z + 68 + 59 = 180, so z = 53º. small error leading to one incorrect answer three correct answers, work shown two correct answers, work shown PTS: 1 DIF: L3 REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem OBJ: 3-4.2 Using Exterior Angles of Triangles NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4b KEY: Triangle Angle-Sum Theorem | vertical angles | supplementary angles | extended response | rubric-based question OTHER 10. ANS: l and m are both perpendicular to n. Explanation: Since l and m are parallel, ∠1 and ∠2 are supplementary by the Same-Side Interior Angles Theorem. It is given that m∠1 = m∠2, so 180 = m∠1 + m∠2 = m∠1 + m∠1 = 2m∠1, and m∠1 = 90 = m∠2. Since ∠1 and ∠2 are right angles, l is perpendicular to n and m is perpendicular to n. PTS: OBJ: STA: KEY: 1 DIF: L3 REF: 3-1 Properties of Parallel Lines 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1 NJ 4.1.12 B.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a perpendicular lines | reasoning | writing in math 3 ID: A 11. ANS: m∠1 + m∠2 + m∠3 = 180. Given m∠1 + m∠2 = m∠3, by substitution, m∠3 + m∠3 = 180. 2m∠3 = 180, and m∠3 = 90. Thus, ∠3 is a right angle, and the triangle is a right triangle. PTS: 1 DIF: L4 REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem OBJ: 3-4.1 Finding Angle Measures in Triangles NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4b KEY: Triangle Angle-Sum Theorem | reasoning | writing in math 12. ANS: 4 ; Line r is perpendicular to line p because a line perpendicular to one of two parallel lines is also 9 perpendicular to the other. Thus, the slope of line r is the opposite reciprocal of the slope of line p. PTS: 1 DIF: L3 REF: 3-7 Slopes of Parallel and Perpendicular Lines OBJ: 3-7.2 Slope and Perpendicular Lines NAT: NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 C.1b | NJ 4.2.12 C.1d | NJ 4.2.12 C.1e | NJ 4.3.12 C.1a | NJ 4.3.12 C.2 KEY: perpendicular lines | parallel lines | slopes of parallel lines | slopes of perpendicular lines | reasoning | writing in math 4