6-2 Parallelograms. Refer to Page 407. a. If. b. If. c. If MQ = 4, what is NP? ANSWER: a. 148 b. 125 c. 4
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1 1. NAVIGATION To chart a course, sailors use a parallel ruler. One edge of the ruler is placed along the line representing the direction of the course to be taken. Then the other ruler is moved until its edge reaches the compass rose printed on the chart. Reading the compass determines which direction to travel. The rulers and the crossbars of the tool form Refer to Page 407. a. If. b. If. c. If MQ = 4, what is NP? a. 148 b. 125 c. 4 ALGEBRA Find the value of each variable in each parallelogram a = 9, b = 8 w = 5, b = 4 6. COORDINATE GEOMETRY Determine the coordinates of the intersection of the diagonals of with vertices A( 4, 6), B(5, 6), C(4, 2), and D( 5, 2). (0, 2) esolutions Manual - Powered by Cognero Page 1
2 JUSTIFY ARGUMENTS Write the indicated type of proof. 7. paragraph is a right angle. are right angles. (Theorem 6.6) 8. two-column ABCH and DCGF are parallelograms. 6.6) is a right angle. are right angles. (Theorem By definition of a parallelogram,. Since is a right angle,. By the Perpendicular Transversal Theorem,. is a right angle, because perpendicular lines form a right angle. because opposite angles in a parallelogram are congruent. are right angles, since all right angles are congruent. ABCH and DCGF are parallelograms. 1. ABCH and DCGF are parallelograms.(given) 2. (Vert. ) 3. (Opp. ) 4. (Substitution.) Use to find each measure QR QP 5 esolutions Manual - Powered by Cognero Page 2
3 APPLY MATH Wesley is a member of the kennel club in his area. His club uses accordion fencing like the section shown at the right to block out areas at dog shows. 13. HOME DECOR The slats on Venetian blinds are designed to remain parallel in order to direct the path of light coming in a widow. In Find each measure. and. a. Identify two pairs of congruent segments. b. Identify two pairs of supplementary angles. a. b. Sample answer: a. JH b. GH c. d. a. 1 in. b. c. 62 d ALGEBRA Find the value of each variable in each parallelogram. a = 7, b = x = 79, y = 101 esolutions Manual - Powered by Cognero Page 3
4 22. W( 4, 5), X(5, 7), Y(4, 2), Z( 5, 4) (0, 1.5) 17. x = 5, y =17 PROOF Write a two-column proof. 23. WXTV and ZYVT are parallelograms. 18. a = 2, b = 3 WXTV and ZYVT are parallelograms. 19. x = 58, y = WXTV and ZYVT are parallelograms. (Given) 2. (Opp. Sides of a.) 3. (Trans. Prop.) 20. z = 2, y = 5 COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of with the given vertices. 21. W( 1, 7), X(8, 7), Y(6, 2), Z( 3, 2) (2.5, 2.5) esolutions Manual - Powered by Cognero Page 4
5 FLAGS Refer to the Alabama state flag at the right. 2. (Opp..) 3. (Isos. Thm.) 4. (Trans. Prop.) : 2. (CPCTC) 3. (Vert..) 4. (CPCTC) 5. (AAS) 6. (CPCTC) JUSTIFY ARGUMENTS Write the indicated type of proof. 26. two-column and are supplementary. (Theorem 6.5) 2. (Opp. sides of a.) 3. are supplementary, are supplementary, are supplementary, and are supplementary. (Cons. int. are suppl.) esolutions Manual - Powered by Cognero Page 5
6 27. two-column (Theorem 6.8) 29. paragraph (Theorem 6.7) is a parallelogram. 2. (Opp. sides of a.) 3. (Opp..) 4. (SAS) 28. two-column (Theorem 6.3) 2. Draw an auxiliary segment and label angles 1, 2, 3, and 4 as shown. (Diagonal of PQRS) 3. (Opp. sides of a.) 4. (Alt. int. Thm.) 5. (Refl. Prop.) 6. (ASA) 7. (CPCTC) It is given that ACDE is a parallelogram. Since opposite sides of a parallelogram are congruent,. By definition of a parallelogram,. because alternate interior angles are congruent. by ASA. CPCTC. By the definition of segment bisector, 30. COORDINATE GEOMETRY Use the graph shown. a. Use the Distance Formula to determine if the diagonals of JKLM bisect each other. Explain. b. Determine whether the diagonals are congruent. Explain. c. Use slopes to determine if the consecutive sides are perpendicular. Explain. a. ; since JP = LP and MP = KP, the diagonals bisect each other. b. No; JP + LP MP + KP. c. No; the slope of and the slope of. The slopes are not negative reciprocals of each other. by esolutions Manual - Powered by Cognero Page 6
7 ALGEBRA Use or value. to find each measure 31. x y COORDINATE GEOMETRY has vertices A( 3, 5), B(1, 2), and C(3, 4). Determine the coordinates of vertex D if it is located in Quadrant III. ( 1, 1) 38. MECHANICS Scissor lifts are variable elevation work platforms. In the diagram, ABCD and DEFG are congruent parallelograms. a. List the angle(s) congruent to. Explain your reasoning. b. List the segment(s) congruent to. Explain your reasoning. c. List the angle(s) supplementary to. Explain your reasoning. a. sample answer: is congruent to because opposite angles of parallelograms are congruent. is congruent to because the parallelograms are congruent, and is congruent to because opposite angles of parallelograms are congruent and congruent to by the Transitive Property. b. ; sample answer: is congruent to because opposite sides of parallelograms are congruent. is congruent to because the parallelograms are congruent, and to is congruent because opposite sides of parallelograms are congruent and congruent to by the Transitive Property. c. sample answer: Angles ABC and ADC are supplementary to because consecutive angles of parallelograms are supplementary. is supplementary to because it is congruent to by the Vertical angles Theorem and supplementary to by substitution. is congruent to because opposite angles of parallelograms are congruent and it is supplementary to by substitution esolutions Manual - Powered by Cognero Page 7
8 PROOF Write a two-column proof (Opp..) 3. (Opp. sides of a.) 4. are rt. ( lines form rt..) 5. are rt. triangles. (Def. of rt. triangles) 6. (HA) 40. MULTIPLE REPRESENTATIONS In this problem, you will explore tests for parallelograms. a. GEOMETRIC Draw three pairs of segments that are both congruent and parallel and connect the endpoints to form quadrilaterals. Label one quadrilateral ABCD, one MNOP, and one WXYZ. Measure and label the sides and angles of the quadrilaterals. b. TABULAR Complete the table below for each quadrilateral. c. VERBAL Make a conjecture about quadrilaterals with one pair of segments that are both congruent and parallel. a. Sample answer: b. c. Sample answer: If a quadrilateral has a pair of sides that are, then the quadrilateral is a parallelogram. esolutions Manual - Powered by Cognero Page 8
9 41. ANALYZE RELATIONSHIPS ABCD is a parallelogram with side lengths as indicated in the figure. The perimeter of ABCD is 22. Find AB. 44. JUSTIFY ARGUMENTS Find in the figure. Explain WRITING IN MATH Explain why parallelograms are always quadrilaterals, but quadrilaterals are sometimes parallelograms. A parallelogram is a polygon with four sides in which the opposite sides and angles are congruent. Quadrilaterals are defined as four-sided polygons. Since a parallelogram always has four sides, it is always a quadrilateral. A quadrilateral is only a parallelogram when the opposite sides and angles of the polygon are congruent. 43. ORGANIZE IDEAS Provide a counterexample to show that parallelograms are not always congruent if their corresponding sides are congruent. m 1 = 116, m 10 = 115; sample answer: m 8 = 64 because alternate interior angles are congruent. 1 is supplementary to 8 because consecutive angles in a parallelogram are supplementary, so m 1 is is supplementary to the 65 degree angle because consecutive angles in a parallelogram are supplementary, so m 10 is or WRITING IN MATH Summarize the properties of the sides, angles, and diagonals of a parallelogram. Sample answer: In a parallelogram, the opp. sides and.two consecutive supplementary. If one angle of a is right, then all the angles are right. The diagonals of a parallelogram bisect each other. 46. The park near Karla s home is a parallelogram, as shown in the figure. Side is 3 times as long as side, and the perimeter of the park is 1200 feet. What is the length of? A 450 ft B 400 ft C 300 ft D 150 ft D esolutions Manual - Powered by Cognero Page 9
10 47. In parallelogram ABCD, A and B are consecutive angles. The measure of A is 40 more than the measure of B. What is the measure of A? F 65 G 70 H 110 J 140 H 48. ACT/SAT In the figure, quadrilateral MNPQ is a parallelogram. Which of the following statements about the figure must be true? I. II. III. A I only B II only C III only D I and II only E I, II, and III D 49. What is the x-coordinate of the intersection of the diagonals of a parallelogram with vertices A( 5, 0), B (3, 4), C(6, 3), and D( 2, 1)? 0.5 esolutions Manual - Powered by Cognero Page 10
8-2 Parallelograms. Refer to Page 489. a. If. b. If c. If MQ = 4, what is NP? SOLUTION:
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