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6-2. Properties of Parallelograms. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Are you ready? Find the value of each variable. 1. x 2. y 3. z. Objectives. TSW prove and apply properties of parallelograms. TSW use properties of parallelograms to solve problems.

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  1. 6-2 Properties of Parallelograms Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Are you ready? Find the value of each variable. 1.x2.y 3.z

  3. Objectives TSW prove and apply properties of parallelograms. TSW use properties of parallelograms to solve problems.

  4. Vocabulary parallelogram

  5. Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.

  6. Helpful Hint Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.

  7. A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .

  8. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find CF. Example 1: Properties of Parallelograms

  9. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Example 2: Properties of Parallelograms

  10. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find DF. Example 3: Properties of Parallelograms

  11. Example 4 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find KN.

  12. Example 5 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find mNML. Def. of angles.

  13. Example 6 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find LO.

  14. Example 7: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ.

  15. Example 8: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find mZ.

  16. Example 9 EFGH is a parallelogram. Find JG.

  17. Example 10 EFGH is a parallelogram. Find FH.

  18. Remember! When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices.

  19. L K J Example 11: Parallelograms in the Coordinate Plane Three vertices of JKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M. Since JKLM is a parallelogram, both pairs of opposite sides must be parallel.

  20. Q S P Example 12 Three vertices of PQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R. Since PQRS is a parallelogram, both pairs of opposite sides must be parallel.

  21. 6.3 Are you ready? Justify each statement. 1. 2. Evaluate each expression for x = 12 and y = 8.5. 3. 2x + 7 4. 16x –9 5.(8y + 5)°

  22. Objective TSW prove that a given quadrilateral is a parallelogram.

  23. You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you can use the definition of a parallelogram or the conditions below.

  24. The two theorems below can also be used to show that a given quadrilateral is a parallelogram.

  25. Example 1: Verifying Figures are Parallelograms Show that JKLM is a parallelogram for a = 3 and b = 9.

  26. Example 2: Verifying Figures are Parallelograms Show that PQRS is a parallelogram for x = 10 and y = 6.5.

  27. Example 3 Show that PQRS is a parallelogram for a = 2.4 and b = 9.

  28. Example 4: Applying Conditions for Parallelograms Determine if the quadrilateral must be a parallelogram. Justify your answer.

  29. Example 5: Applying Conditions for Parallelograms Determine if the quadrilateral must be a parallelogram. Justify your answer.

  30. Example 6 Determine if the quadrilateral must be a parallelogram. Justify your answer.

  31. Example 7 Determine if each quadrilateral must be a parallelogram. Justify your answer.

  32. Helpful Hint To say that a quadrilateral is a parallelogram by definition, you must show that both pairs of opposite sides are parallel.

  33. Example 8: Proving Parallelograms in the Coordinate Plane Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0).

  34. Example 9: Proving Parallelograms in the Coordinate Plane Show that quadrilateral ABCD is a parallelogram by using Theorem 6-3-1.A(2, 3), B(6, 2), C(5, 0), D(1, 1).

  35. Example 10 Use the definition of a parallelogram to show that the quadrilateral with vertices K(–3, 0), L(–5, 7), M(3, 5), and N(5, –2) is a parallelogram.

  36. You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply.

  37. Helpful Hint To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions.

  38. Example 11: Application The legs of a keyboard tray are connected by a bolt at their midpoints, which allows the tray to be raised or lowered. Why is PQRS always a parallelogram?

  39. Example 12 The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same?

  40. Lesson Quiz: Part I 1. Show that JKLM is a parallelogram for a = 4 and b = 5. 2. Determine if QWRT must be a parallelogram. Justify your answer. JN = LN = 22; KN = MN = 10; so JKLM is a parallelogram by Theorem 6-3-5. No; One pair of consecutive s are , and one pair of opposite sides are ||. The conditions for a parallelogram are not met.

  41. Lesson Quiz: Part II 3. Show that the quadrilateral with vertices E(–1, 5), F(2, 4), G(0, –3), and H(–3, –2) is a parallelogram. Since one pair of opposite sides are || and , EFGH is a parallelogram by Theorem 6-3-1.

  42. Step 4 Use the slope formula to verify that R Q S P Check It Out! Example 3 Continued The coordinates of vertex R are (7, 6).

  43. Lesson Quiz: Part I In PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure. 1.PW2. mPNW 18 144°

  44. Lesson Quiz: Part II QRST is a parallelogram. Find each measure. 2.TQ3. mT 71° 28

  45. Lesson Quiz: Part III 5. Three vertices of ABCD are A (2, –6), B (–1, 2), and C(5, 3). Find the coordinates of vertex D. (8, –5)

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