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Name ________________________________________ Date ___________________ Class __________________ LESSON 6-3 Practice A Conditions for Parallelograms For each statement, tell what information you would need to show about the figure to conclude that the figure is a parallelogram. R List objects used with ,║, or connecting them: Name of Formula Needed: Numbers Should Be: If both pairs of opposite sides of a 1. quadrilateral are parallel, then the 1.quadrilateral is a parallelogram. 1st pair of opposite sides: > > 2nd pair of opposite sides: > > If oIf If one pair of opposite sides of a 2. quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. Pair of opposite sides: > > > > If the diagonals of a quadrilateral 3. bisect each other, then the quadrilateral is a parallelogram. Halves of 1st diagonal: > > Halves of 2nd diagonal: > > 1st pair of opposite sides: > > 2nd pair of opposite sides: > > Statement: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram A quadrilateral has vertices E(1, 1), F(4, 5), G(6, 6), H(3, 2). Complete Exercises 7–10 to tell whether EFGH is a parallelogram. 5. Plot the vertices and draw EFGH. 6. Use the Distance Formula: EF ________ HG ________ 7. Use the Slope Formula: slope of EF ________ slope of HG ________ 8. The answers to Exercises 6 and 7 prove that EFGH is a parallelogram. Which one of Exercises 1–4 states the theorem that you used? ________ This desk lamp has a circular base and a movable arm in the shape of a parallelogram. Use the figure to answer Exercises 11–13. 10. AD is vertical. Name another side of parallelogram ABCD that is also vertical. ________ 11. Because AD is attached to the base, AD stays vertical as the arm is moved. Make a conjecture about what happens to BC as the arm is moved up or down. ________________________________________________________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A57 Holt Geometry 7. PQ RS 26 ; QR PS 5 2 ; PQRS is a parallelogram. LESSON 6-3 Practice A 8. Possible answer: UV TW 2 5 ; slope of UV slope of TW 2; TUVW is a parallelogram. Practice C 1. W Y and X Z 2. WX ZY and WZ XY 3. Possible answer: W is supplementary to X and to Z. 1. A(4, 4), B(2, 5), C(2, 5), D(0, 26) 2. Possible answer: A C 4. Possible answer: WX ZY and WX ZY 3. AB || CD ; possible answer: because A B and C D and the sum of the interior angle measures of a quadrilateral is 360°, 2mA 2mD 360° or 2(mA mD) 360°. Therefore mA mD 180°. A and D are supplementary, so by the Converse of the Same-Side Interior Angles Theorem, AB || CD 5. WY and XZ bisect each other. 6. WX ZY and WZ XY 7. 8. 5; 5 10. 4 9. 4. All four sides are congruent, and the two pairs of opposite angles are congruent; possible answer: because the diagonals are perpendicular, all four angles created by the intersecting diagonals are right angles and therefore congruent. And because the diagonals bisect each other, all four of the right triangles are congruent by SAS. By CPCTC, all four of the parallelogram’s sides must be congruent. The two pairs of opposite angles are congruent as for any parallelogram. 4 4 ; 3 3 11. BC 12. BC moves up or down but stays vertical. Practice B 1. ABCD is a parallelogram. mA mC 72 and mB mD 108 EFGH is not a parallelogram. HI 8.6 and FI 7.6. EG does not bisect HF . 3. No, the diagonals do not necessarily bisect each other. 5. 4. Yes, the triangles with numbered angles are by AAS. By CPCTC, the parallel sides are congruent. 5. No, x x may not be 180. 6. slope of JK slope of LM 1; slope 2 of KL slope of JM 3 ; JKLM is a parallelogram. All four sides are congruent, and all four angles are congruent; possible answer: the sides are congruent for the same reasons given in Exercise 4. But because the diagonals are congruent and bisected, each right triangle.created by the diagonals is an isosceles right triangle. The acute angles of these triangles have measure 45°, so all the angles of the Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A57 Holt Geometry Name _______________________________________ Date ___________________ Class __________________ parallelogram have measure 90°. 6. 90° Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-20 Holt Geometry