8.3 show that a quadrilateral is a parallelogram

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8.3 show that a quadrilateral is a parallelogram

  1. 1. 8.3 Show that a Quadrilateral is a Parallelogram 8.3 Bell Thinger 1. What congruence postulate shows that ABE CDE ? ANSWER 2. If E ANSWER SAS G, find m 118º E.
  2. 2. 8.3
  3. 3. 8.3 Example 1 Ride An amusement park ride has a moving platform attached to four swinging arms. The platform swings back and forth, higher and higher, until it goes over the top and around in a circular motion. In the diagram below, AD and BC represent two of the swinging arms, and DC is parallel to the ground (line l). Explain why the moving platform AB is always parallel to the ground.
  4. 4. 8.3 Example 1 SOLUTION The shape of quadrilateral ABCD changes as the moving platform swings around, but its side lengths do not change. Both pairs of opposite sides are congruent, so ABCD is a parallelogram by Theorem 8.7. By the definition of a parallelogram, AB DC . Because DC is parallel to line l, AB is also parallel to line l by the Transitive Property of Parallel Lines. So, the moving platform is parallel to the ground.
  5. 5. 8.3 Guided Practice 1. In quadrilateral WXYZ, m W = 42°, m X = 138°, m Y = 42°. Find m Z. Is WXYZ a parallelogram? Explain your reasoning. ANSWER 138°; yes; the sum of the measures of the interior angles in a quadrilateral is 360°, so the measure of Z is 138°. Since opposite angles of the quadrilateral are congruent, WXYZ is a parallelogram.
  6. 6. 8.3
  7. 7. 8.3 Example 2 ARCHITECTURE The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU. SOLUTION In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram. By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.
  8. 8. 8.3 Example 3 ALGEBRA For what value of x is quadrilateral CDEF a parallelogram? SOLUTION By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN . Find x so that FN DN .
  9. 9. 8.3 Example 3 FN = DN 5x – 8 = 3x 2x – 8 = 0 2x = 8 x= 4 Set the segment lengths equal. Substitute 5x –8 for FN and 3x for DN. Subtract 3x from each side. Add 8 to each side. Divide each side by 2. When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12. Quadrilateral CDEF is a parallelogram when x = 4.
  10. 10. 8.3 Guided Practice What theorem can you use to show that the quadrilateral is a parallelogram? 2. ANSWER
  11. 11. 8.3 Guided Practice What theorem can you use to show that the quadrilateral is a parallelogram? 3. ANSWER
  12. 12. 8.3 Guided Practice What theorem can you use to show that the quadrilateral is a parallelogram? 4. ANSWER
  13. 13. 8.3 Guided Practice 5. For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. ANSWER 2; The diagonals of a parallelogram bisect each other so solve 2x = 10 – 3x for x.
  14. 14. 8.3 Example 4 Show that quadrilateral ABCD is a parallelogram. SOLUTION One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9. First use the Distance Formula to show that AB and CD are congruent. AB = [2 – (–3)]2 + (5 – 3)2 = 29 CD = (5 – 0)2 + (2 – 0)2 29 Because AB = CD = = 29 , AB CD.
  15. 15. 8.3 Example 4 Then use the slope formula to show that AB CD. 5 – (3) 2 Slope of AB = 2 – (–3) = 5 2–0 2 Slope of CD = 5 – 0 = 5 Because AB and CD have the same slope, they are parallel. AB and CD are congruent and parallel. So, ABCD is a parallelogram by Theorem 8.9.
  16. 16. 8.3
  17. 17. Exit 8.3 Slip Tell how you know that the quadrilateral is a parallelogram 1. ANSWER The diagonals bisect each other.
  18. 18. Exit 8.3 Slip Tell how you know that the quadrilateral is a parallelogram 2. ANSWER Both pairs of opposite angles congruent. You can also conclude that opposite sides are parallel and use the definition of a parallelogram.
  19. 19. Exit 8.3 Slip 3. For what value of x is the quadrilateral a parallelogram. ANSWER 50
  20. 20. Exit 8.3 Slip 4. In a coordinate plane, draw the quadrilateral ABCD with vertices A(–1, 3), B(7, –1), C(6, –4), and D(–2, 0). show that ABCD is a parallelogram. ANSWER AB = CD = 4 5 and BC = AD = 10, so two pairs of opposite sides are congruent.
  21. 21. 8.3 Pg 546-549 #6, 9, 19, 25, 30

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