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Parallelograms

Parallelograms. 5-1. ALGEBRA. Find the values of x and y . CD. AB =. Opposite sides of a are . 12. x + 4 =. x =. 8. or m A = m C . . By Theorem 8.4, A C , . ANSWER. In ABCD , x = 8 and y = 65. EXAMPLE 1. Use properties of parallelograms.

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Parallelograms

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  1. Parallelograms 5-1

  2. ALGEBRA Find the values of xand y. CD AB= Opposite sides of a are . 12 x + 4= x = 8 or m A = m C. By Theorem 8.4, A C, ANSWER InABCD, x = 8andy = 65. EXAMPLE 1 Use properties of parallelograms ABCDis a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. Substitute x + 4 for ABand 12 for CD. Subtract 4 from each side. So, y ° = 65°.

  3. Find FGand m G. 1. FG = HE x = 8 Opposite sides of a are . or m E = m G. By Theorem 8.4, E G, ANSWER InFEHG, FG = 8andm G = 60°. for Example 1 GUIDED PRACTICE SOLUTION So, G ° = 60°.

  4. Find the values of xand y. 2. JK = ML 18 = y + 3 15 = y Opposite sides of a are . or m J = m L. 2x = 50 x = 25 By Theorem 8.4, J L, ANSWER InJKLM, x = 25andy = 15. for Example 1 GUIDED PRACTICE SOLUTION Substitute 18 for JKand y + 3 for ML. Subtract 3 from each side. Substitute Divide 2 from each side.

  5. Desk Lamp As shown, part of the extending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. Find m BCDwhen m ADC = 110°. By Theorem 8.5, the consecutive angle pairs in ABCD are supplementary. So, m ADC + m BCD = 180°. Because m ADC = 110°, m BCD =180° –110° = 70°. EXAMPLE 2 Use properties of parallelograms SOLUTION

  6. By Theorem 8 .6, the diagonals of a parallelogram bisect each other. So, Pis the midpoint of diagonals LNand OM. Use the Midpoint Formula. 4 + 0 7 7 + 0 , , 2 Coordinates of midpoint Pof OM = ( ) ( ) = 2 2 2 ANSWER The correct answer is A. Standardized Test Practice EXAMPLE 3 SOLUTION

  7. Find the indicated measure in JKLM. 3. NM By Theorem 8 .6, the diagonals of a parallelogram bisect each other. So, Nis the midpoint of diagonals KM . KN = NM 2 = NM for Examples 2 and 3 GUIDED PRACTICE SOLUTION Substitute

  8. Find the indicated measure in JKLM. 4. KM KM = KN + NM KM = KM = for Examples 2 and 3 GUIDED PRACTICE SOLUTION By theorem 8.6 2 + 2 Substitute 4 Add

  9. Find the indicated measure in JKLM. 5. m JML By Theorem 8.5, the consecutive angle pairs in JKLM are supplementary. So, m KJM + m JML = 180°. Because m KJM = 110°, m JML =180° –110° = 70°. for Examples 2 and 3 GUIDED PRACTICE SOLUTION

  10. Find the indicated measure in JKLM. 6. m KML m JML = m KMJ + m KNL 30° + m KML m KML for Examples 2 and 3 GUIDED PRACTICE SOLUTION 70° = Substitute 40° = Subtract

  11. 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.

  12. 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

  13. 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,

  14. 2. ANSWER In the graphic, two opposite sides are equal, i.e, 30m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  15. 3. ANSWER Two pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram. By theorem 8.7 for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  16. 4. ANSWER By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  17. 5. For what value of xis quadrilateral MNPQa parallelogram? Explain your reasoning. 2x = 10 – 3x By Theorem 8.6 [ Diagonals in bisect each other ] 5x = 10 x = 2 for Examples 2 and 3 GUIDED PRACTICE SOLUTION Add 3xto each side Divide each side by 5

  18. 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.

  19. 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

  20. 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,

  21. 2. ANSWER In the graphic, two opposite sides are equal, i.e, 30m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  22. 3. ANSWER Two pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram. By theorem 8.7 for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  23. 4. ANSWER By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram. for Examples 2 and 3 GUIDED PRACTICE What theorem can you use to show that the quadrilateral is a parallelogram?

  24. 5. For what value of xis quadrilateral MNPQa parallelogram? Explain your reasoning. 2x = 10 – 3x By Theorem 8.6 [ Diagonals in bisect each other ] 5x = 10 x = 2 for Examples 2 and 3 GUIDED PRACTICE SOLUTION Add 3xto each side Divide each side by 5

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