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Name: Date: Period: . Practice A 3.1: Angles Formed by Parallel Lines and Transversals 1. The Corresponding Angles Postulate states that if two parallel lines are cut by a transversal, then the pairs of corresponding angles are ____________________. 2. Congruent angles have _____________________ measures. Find each angle measure. 3.m1 _______________________ 4. m2 _______________________ Find x. 5. 6. ________________________________________ ________________________________________ Fill in the blanks to complete these theorems about angle pairs. 7. If two _____________________ lines are cut by a _____________________, then the two pairs of alternate interior angles are congruent. 8. If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are _____________________. 9. If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are _____________________. Practice B 3.1: Angles Formed by Parallel Lines and Transversals 1. m1 _______________________ 2. m2 _______________________ Find each angle measure. 3. mABC _______________________ 4. mDEF _______________________ Complete the two-column proof, using the Proof Bank, to show that same-side exterior angles are supplementary. Hint: Draw a picture first. 5. Given: p || q Proof Bank: Prove: m1 m3 180 Proof: Subst. Statements Reasons 1. p || q 1. Given 2. a. _______________________ 2. Lin. Pair Thm. Corr. s Post. m2 + m3 = 180° m1 = m2 m1 + m3 = 180° 3. 1 2 3. b. _______________________ 4. c. _______________________ 4. Def. of s 5. d. _______________________ 5. e. _______________________ Problem Solving 3.1: Angles Formed by Parallel Lines and Transversals Find each value. Name the postulate or theorem that you used to find the values. 1. In the diagram of movie theater seats, the incline of the floor, f, is parallel to the seats, s. If m168, what is x? ________________________________________ 2. In the diagram, roads a and b are parallel. What is the measure of PQR? ________________________________________ Use the diagram of a staircase railing for Exercises 4 and 5. AG || CJ and AD || FJ . Choose the best answer. 3. Which is a true statement about the measure of DCJ? A It equals 30, by the Alternate Interior Angles Theorem. B It equals 30, by the Corresponding Angles Postulate. C It equals 50, by the Alternate Interior Angles Theorem. D It equals 50, by the Corresponding Angles Postulate. 4.Which is a true statement about the value of n? F It equals 25, by the Alternate Interior Angles Theorem. G It equals 25, by the Same-Side Interior Angles Theorem. H It equals 35, by the Alternate Interior Angles Theorem. J It equals 35, by the Same-Side Interior Angles Theorem. Practice A 3.2 Proving Lines Parallel 1. The Converse of the Corresponding Angles Postulate states that if two coplanar lines are cut by a transversal so that a pair of corresponding angles is congruent, then the two lines are ____________________. Use the figure for Exercises 2 and 3. Given the information in each exercise, state the reason why lines b and c are parallel. 2. 4 8 3. m368, m7(5x3), x13 ________________________________________ _________________________________________ ________________________________________ _________________________________________ Fill in the blanks to complete these theorems about parallel lines. 4. If two coplanar lines are cut by a ______________________ so that a pair of alternate interior angles are ______________________, then the two lines are parallel. 5. If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are ______________________, then the two lines are parallel. 6. If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are ______________________. 7. Shu believes that a theorem is missing from the lesson. His conjecture is that if two coplanar lines are cut by a transversal so that a pair of same-side exterior angles are supplementary, then the two lines are parallel. Complete the two-column proof with the statements and reasons provided. m || n 2 and 3 are supplementary. Given Supps. Thm. Given: 1 and 3 are supplementary. Prove: m || n Proof: Statements Reasons 1. 1 and 3 are supplementary. 1. a. ___________________________ 2. b. ___________________________ 2. Linear Pair Thm. 3. 1 2 3. c. ___________________________ 4. d. ___________________________ 4. Conv. of Corr. s Post. Problem Solving 3.2 Proving Lines Parallel 1. A bedroom has sloping ceilings as shown. Marcel is hanging a shelf below a rafter. If m1(8x 1), m2(6x7), and x4, show that the shelf is parallel to the rafter above it. 2. In the sign, m3(3y7), m4(5y5), and y21. Show that the sign posts are parallel. Choose the best answer. 3. In the bench, mEFG(4n16), mFJL(3n40), mGKL(3n22), and n24. Which is a true statement? A FG || HK by the Converse of the Corr. s Post. B FG || HK by the Converse of the Alt. Int. s Thm. C EJ || GK by the Converse of the Corr. s Post. D EJ || GK by the Converse of the Alt. Int. s Thm. 4.In the windsurfing sail, m5(7c1), m6(9c 1), m717c, and c6. Which is a true statement? F RV is parallel to SW . G SW is parallel to TX . H RT is parallel to VX . J Cannot conclude that two segments are parallel The figure shows Natalia’s initials, which are monogrammed on her duffel bag. Use the figure for Exercises5and 6. 5. If m1(4x 24), m2(2x8), and x16, show that the sides of the letter N are parallel. 6. If m3(7x13), m4(5x35), and x11, show that the sides of the letter H are parallel. ________________________________________ _________________________________________ Practice A 3.3 Perpendicular Lines 1. The perpendicular bisector of a segment is a line ______________________ to a segment at the segment’s ______________________. 2. The shortest segment from a point to a line is ______________________ to the line. For Exercises 3 and 4, name the shortest segment from the point to the line and write an inequality for x. 3. 4. ________________________________________ ________________________________________ Fill in the blanks to complete these theorems about parallel and perpendicular lines. 5. If two coplanar lines are perpendicular to the same line, then the two lines are ______________________ to each other. 6. If two intersecting lines form a linear pair of ______________________ angles, then the lines are perpendicular. 7. In a plane, if a transversal is perpendicular to one of two parallel lines, then it is ______________________ to the other line. Practice B 3.3 Perpendicular Lines For Exercises 1–4, name the shortest segment from the point to the line and write an inequality for x. 1. 2. ________________________________________ ________________________________________ 3. ________________________________________ Complete the two-column proof using the Proof Bank. 4. Given: m n Prove: 1 and 2 are a linear pair of congruent angles. Proof Bank: Proof: Statements Reasons 1. a. ___________________________ 1. Given 2. b. ___________________________ 2. Def. of 3. 1 2 3. c. ___________________________ 4. m1m2180 4. Add. Prop. of 5 5. d. ___________________________ 5. Def. of linear pair 1 and 2 are a linear pair. Def. of s m1 90°, m2 90° mn