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Name ________________________________________ Date ___________________ Class __________________ LESSON 6-3 Reteach Conditions for Parallelograms You can use the following conditions to determine whether a quadrilateral such as PQRS is a parallelogram. Conditions for Parallelograms QR ≅ SP QR & SP QR ≅ SP If one pair of opposite sides is and ≅, then PQRS is a parallelogram. . ∠P ≅ ∠R ∠Q ≅ ∠S If both pairs of opposite angles are ≅, then PQRS is a parallelogram. PQ ≅ RS If both pairs of opposite sides are ≅, then PQRS is a parallelogram. PT ≅ RT QT ≅ ST If the diagonals bisect each other, then PQRS is a parallelogram. A quadrilateral is also a parallelogram if one of the angles is supplementary to both of its consecutive angles. 65° + 115° = 180°, so ∠A is supplementary to ∠B and ∠D. Therefore, ABCD is a parallelogram. Show that each quadrilateral is a parallelogram for the given values. Explain. 1. Given: x = 9 and y = 4 2. Given: w = 3 and z = 31 _________________________________________ ________________________________________ _________________________________________ ________________________________________ _________________________________________ ________________________________________ _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-22 Holt McDougal Geometry Name ________________________________________ Date ___________________ Class __________________ LESSON 6-3 Reteach Conditions for Parallelograms continued You can show that a quadrilateral is a parallelogram by using any of the conditions listed below. Conditions for Parallelograms • Both pairs of opposite sides are parallel (definition). • One pair of opposite sides is parallel and congruent. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • One angle is supplementary to both its consecutive angles. EFGH must be a parallelogram because both pairs of opposite sides are congruent. JKLM may not be a parallelogram because none of the sets of conditions for a parallelogram is met. Determine whether each quadrilateral must be a parallelogram. Justify your answer. 4. 3. _________________________________________ ________________________________________ _________________________________________ ________________________________________ 6. 5. _________________________________________ ________________________________________ _________________________________________ ________________________________________ Show that the quadrilateral with the given vertices is a parallelogram by using the given definition or theorem. 7. J(−2, −2), K(−3, 3), L(1, 5), M(2, 0) Both pairs of opposite sides are parallel. 8. N(5, 1), P(2, 7), Q(6, 9), R(9, 3) Both pairs of opposite sides are congruent. _________________________________________ ________________________________________ _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-23 Holt McDougal Geometry 5. No, x° + x° may not be 180°. triangles have measure 45°, so all the angles of the parallelogram have measure 90°. 6. slope of JK = slope of LM = 1; slope of 2 KL = slope of JM = − ; JKLM is a 3 parallelogram. 7. PQ = RS = 26 ; QR = PS = 5 2 ; PQRS is a parallelogram. 8. Possible answer: UV = TW = 2 5 ; slope of UV = slope of TW = 2; TUVW is a parallelogram. 6. 90° Reteach 1. QR = ST = 12; RS = TQ = 16; both pairs of opp. sides are ≅. Practice C 1. A(4, 4), B(2, 5), C(−2, −5), D(0, 2. DE = FC = 10; m∠E = 118° and m∠F = 62°, so ∠E and ∠F are supp. and DE || FC ; one pair of opposite sides are || and ≅. 6) 2. Possible answer: ∠A ≅ ∠C 3. AB || CD ; possible answer: because ∠A ≅ ∠B and ∠C ≅ ∠D and the sum of the interior angle measures of a quadrilateral is 360°, 2m∠A + 2m∠D = 360° or 2(m∠A + m∠D) = 360°. Therefore m∠A + m∠D = 180°. ∠A and ∠D are supplementary, so by the Converse of the Same-Side Interior Angles Theorem, AB || CD 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. 3. Yes; one pair of opp. sides is || and ≅. 4. Yes; the diagonals bisect each other. 5. No; none of the sets of conditions for a parallelogram is met. 6. Yes; both pairs of opp. ∠ are ≅. S 7. slope of JK = slope of LM = −5; slope of KL = slope of MJ = 1 2 8. NP = QR = 3 5 ; PQ =RN = 2 5 Challenge 1. 2. –11. Arrangements will vary. Problem Solving 1. Yes; both pairs of opposite sides of quadrilateral LMNP remain congruent, so LMNP is always a . 2. 56° . 3. Possible answer: m∠F = 120° 5. 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 4. Possible answer: y = −x + 1; both pairs of opposite sides have the same slope, so they are parallel. 5. C 6. H Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A62 Holt McDougal Geometry