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EXAMPLE 1

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raymondarivera
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EXAMPLE 1

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  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, ADand BCrepresent two of the swinging arms, and DCis parallel to the ground (line l). Explain why the moving platform ABis always parallel to the ground. EXAMPLE 1 Solve a real-world problem

  2. By the definition of a parallelogram, AB DC. Because DCis parallel to line l, ABis also parallel to line l by the Transitive Property of Parallel Lines. So, the moving platform is parallel to the ground. EXAMPLE 1 Solve a real-world problem SOLUTION The shape of quadrilateral ABCDchanges as the moving platform swings around, but its side lengths do not change. Both pairs of opposite sides are congruent, so ABCDis a parallelogram by Theorem 8.7.

  3. 1. In quadrilateral WXYZ, m W = 42°,m X =138°, m Y = 42°. Find m Z. Is WXYZa parallelogram? Explain your reasoning. for Example 1 GUIDED PRACTICE

  4. ARCHITECTURE In the photograph, ST UVand ST UV. By Theorem 8.9, quadrilateral STUVis a parallelogram. EXAMPLE 2 Identify a parallelogram 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 By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.

  5. For what value of xis quadrilateral CDEFa parallelogram? ALGEBRA By Theorem 8.10, if the diagonals of CDEFbisect each other, then it is a parallelogram. You are given that CNEN. Find xso that FN DN. EXAMPLE 3 Use algebra with parallelograms SOLUTION

  6. ANSWER Quadrilateral CDEF is a parallelogram when x = 4. EXAMPLE 3 Use algebra with parallelograms DN FN = Set the segment lengths equal. 3x 5x – 8 = Substitute 5x –8 for FN and 3xfor DN. 0 2x – 8 = Subtract 3xfrom each side. 8 2x = Add 8 to each side. 4 x = Divide each side by 2. FN = 5(4) –8 = 12 andDN = 3(4) = 12. Whenx = 4,

  7. 2. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  8. 3. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  9. What theorem can you use to show that the quadrilateral is a parallelogram? 4. for Examples 2 and 3 GUIDED PRACTICE

  10. 5. For what value of xis quadrilateral MNPQa parallelogram? Explain your reasoning. for Examples 2 and 3 GUIDED PRACTICE

  11. Show that quadrilateral ABCDis a parallelogram. First use the Distance Formula to show that ABand CDare congruent. 29 [2 – (–3)]2 + (5 – 3)2 = 29 (5 – 0)2 + (2 – 0)2 = EXAMPLE 4 Use coordinate geometry SOLUTION One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9. AB = CD =

  12. 29 , AB BecauseAB = CD = CD. AB CD. Then use the slope formula to show that 5 – (3) 2 – 0 Slope of CD = = Slope of AB = = 5 – 0 2 – (–3) Because ABand CDhave the same slope, they are parallel. ANSWER ABand CDare congruent and parallel. So, ABCDis a parallelogram by Theorem 8.9. 2 2 5 5 EXAMPLE 4 Use coordinate geometry

  13. EXAMPLE 4 for Example 4 GUIDED PRACTICE 6. Refer to the Concept Summary. Explain how other methods can be used to show that quadrilateral ABCDin Example 4 is a parallelogram.

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