1 / 24

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.

gautier
Download Presentation

Parallelograms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related