1. Tests for Parallelograms

2. Find the missing measures in parallelogram WXYZ. 1) m<XWZ 2) m<WXY 3) m<WZC 4) m<WCX 5) YZ 6) XZ 7) WY 8) WZ X W YZ C 28 7 9.3 63 42 13.2 9

3. Find the missing measures in parallelogram WXYZ. 1) m<XWZ Alternate interior angles are congruent; 42 = m<ZYW. Opposite angles are congruent. M<ZYX = m<XWZ 42 + 63 = m<XWZ 105 = m<XWZ 2) m<WXY Consecutive angles are supplementary. 180 = m<XWZ + m<WXY 180 = 105 + m<WXY 75 = m<WXY X W YZ C 28 7 9.3 63 42 13.2 9

4. Find the missing measures in parallelogram WXYZ. 3) m<WZC Opposite angles are congruent; m<WXY = m<WZY 75 = m<WZY 75 = m<WZC + m<CZY 75 = m<WZC + 28 47 = m<WZC 4) m<WCX All the angles in a triangle add up to 180. 180 = 42 + 28 + m<WCX 180 = 70 + m<WCX 110 = m<WCX X W YZ C 28 7 9.3 63 42 13.2 9

5. Find the missing measures in parallelogram WXYZ. 5) YZ Opposite sides are congruent. 13.2 = YZ 6) XZ Diagonals bisect each other. CX = 9 = CZ ZX = CX + CZ ZX = 9 + 9 ZX = 18 X W YZ C 28 7 9.3 63 42 13.2 9

6. Find the missing measures in parallelogram WXYZ. 7) WY Diagonals bisect each other. CY = 7 = CW WY = CY + CW WY = 7 + 7 WY = 14 8) WZ Opposite sides are congruent. 9.3 = WZ X W YZ C 28 7 9.3 63 42 13.2 9

7. Theorem 6-5- If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-6- If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-7- If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-8- If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

8. Example 1: The coordinates of the vertices of quadrilateral ABCD are A(-1, 3), B(2, 1), C(9, 2), and D(6, 4). Determine if quadrilateral ABCD is a parallelogram. Find the length of AB and CD. The distance formula is d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) AB d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((-1 – 2) 2 + (3 – 1) 2 ) d=√((-3) 2 + (2) 2 ) d=√(9 + 4) d=√(13) Find the slope of AB and CD. The slope formula is m = (y 2 – y 1 )/(x 2 – x 1 ) CD d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((9 – 6) 2 + (2 – 4) 2 ) d=√((3) 2 + (-2) 2 ) d=√(9 + 4) d=√(13) Opposite sides are congruent and opposite sides are parallel so quadrilateral ABCD is a parallelogram. AB m = (y 2 – y 1 )/(x 2 – x 1 ) m = (3 – 1)/(-1 – 2) m = (2)/(-3) m = -2/3 CD m = (y 2 – y 1 )/(x 2 – x 1 ) m = (2 – 4)/(9 – 6) m = (-2)/(3) m = -2/3

9. Example 2: The coordinates of the vertices of quadrilateral EFGH are E(6, 5), F(6, 11), G(14, 18), and H(14, 12). Determine if quadrilateral ABCD is a parallelogram. Find the length of EF and GH. The distance formula is d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) EF d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((6 – 6) 2 + (5 – 11) 2 ) d=√((0) 2 + (-6) 2 ) d=√(0 + 36) d=√(36) = 6 Find the slope of EF and GH. The slope formula is m = (y 2 – y 1 )/(x 2 – x 1 ) GH d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((14 – 14) 2 + (18 – 12) 2 ) d=√((0) 2 + (6) 2 ) d=√(0 + 36) d=√(36) = 6 Opposite sides are congruent and opposite sides are parallel so quadrilateral ABCD is a parallelogram. EF m = (y 2 – y 1 )/(x 2 – x 1 ) m = (5 – 11)/(6 – 6) m = (-6)/(0) m = undefined GH m = (y 2 – y 1 )/(x 2 – x 1 ) m = (18 – 12)/(14 – 14) m = (6)/(0) m = undefined

10. Example 3: Determine if each quadrilateral is a parallelogram. Justify your answer. A) Yes; Opposite angles are congruent. B) 15 Yes; Diagonals bisect each other. C) 118 No; Not enough information. D) Yes; One pair of opposite sides is parallel and congruent.

11. Example 4: Find the values of x and y that ensure each quadrilateral is a parallelogram. 3x + 17 44x - y 2y For the quadrilateral to be a parallelogram opposite sides must be congruent. Set 4 = 4x – y and solve for y. 4 = 4x – y 4 – 4x = -y 4x – 4 = y Now plug 4x – 4 in for y in the equation 3x + 17 = 2y 3x + 17 = 2y 3x + 17 = 2(4x – 4) 3x + 17 = 8x – 8 17 = 5x – 8 25 = 5x 5 = x Now plug 5 in for x in the equation 4x – 4 = y 4x – 4 = y 4(5) – 4 = y 20 – 4 = y 16 = y

12. Example 5: Find the values of x and y that ensure each quadrilateral is a parallelogram. 64 5y2y + 36 6x - 2 For the quadrilateral to be a parallelogram opposite sides must be congruent. 64 = 6x - 2 66 = 6x 11 = x Now solve for y. 5y = 2y + 36 3y = 36 y = 12

13. Example 6: Find the values of x and y that ensure each quadrilateral is a parallelogram. 7x 4x y + 2 2y For the quadrilateral to be a parallelogram the diagonals bisect each other. Set 4x = y + 2 and solve for y. 4x = y + 2 4x – 2 = y Now plug 4x – 2 in for y in the equation 7x = 2y 7x = 2y 7x = 2(4x – 2) 7x = 8x – 4 -x = -4 x = 4 Now plug 4 in for x in the equation 4x – 2 = y 4x – 2 = y 4(4) – 2 = y 16 – 2 = y 14 = y