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Understanding Parallelograms: Properties & Applications

Learn to prove quadrilaterals as parallelograms, apply rectangle, rhombus, square properties, solve problems, and verify figures.

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Understanding Parallelograms: Properties & Applications

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  1. 6-3 Conditions for Parallelograms Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up Justify each statement. 1. 2. Evaluate each expression for x = 12 and y = 8.5. 3. 2x + 7 4. 16x –9 5.(8y + 5)° Reflex Prop. of  Conv. of Alt. Int. s Thm. 31 183 73°

  3. Objective Prove that a given quadrilateral is a parallelogram. Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems.

  4. Vocabulary rectangle rhombus square

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

  6. 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.

  7. Example 1A: Verifying Figures are Parallelograms Show that JKLM is a parallelogram for a = 3 and b = 9. Step 1 Find JK and LM. Given LM = 10a + 4 JK = 15a – 11 Substitute and simplify. LM = 10(3)+ 4 = 34 JK = 15(3) – 11 = 34

  8. Example 1A Continued Step 2 Find KL and JM. Given KL = 5b + 6 JM = 8b – 21 Substitute and simplify. KL = 5(9) + 6 = 51 JM = 8(9) – 21 = 51 Since JK = LM and KL = JM, JKLM is a parallelogram by Theorem 6-3-2.

  9. Example 1B: Verifying Figures are Parallelograms Show that PQRS is a parallelogram for x = 10 and y = 6.5. mQ = (6y + 7)° Given Substitute 6.5 for y and simplify. mQ = [(6(6.5) + 7)]° = 46° mS = (8y – 6)° Given Substitute 6.5 for y and simplify. mS = [(8(6.5) – 6)]° = 46° mR = (15x – 16)° Given Substitute 10 for x and simplify. mR = [(15(10) – 16)]° = 134°

  10. Example 1B Continued Since 46° + 134° = 180°, R is supplementary to both Q and S. PQRS is a parallelogram by Theorem 6-3-4.

  11. Check It Out! Example 2a Determine if the quadrilateral must be a parallelogram. Justify your answer. Yes The diagonal of the quadrilateral forms 2 triangles. Two angles of one triangle are congruent to two angles of the other triangle, so the third pair of angles are congruent by the Third Angles Theorem. So both pairs of opposite angles of the quadrilateral are congruent . By Theorem 6-3-3, the quadrilateral is a parallelogram.

  12. Check It Out! Example 2b Determine if each quadrilateral must be a parallelogram. Justify your answer. No. Two pairs of consective sides are congruent. None of the sets of conditions for a parallelogram are met.

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

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

  15. Example 4: 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? Since the bolt is at the midpoint of both legs, PE = ER and SE = EQ. So the diagonals of PQRS bisect each other, and by Theorem 6-3-5, PQRS is always a parallelogram.

  16. A second type of special quadrilateral is a rectangle. A rectangleis a quadrilateral with four right angles.

  17. Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.

  18.  diags. bisect each other Example 1: Craft Application A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM. Rect.  diags.  KM = JL = 86 Def. of  segs. Substitute and simplify.

  19. A rhombus is another special quadrilateral. A rhombusis a quadrilateral with four congruent sides.

  20. Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.

  21. Example 2B Continued Rhombus  each diag. bisects opp. s mVTZ =mZTX mVTZ =(5a – 5)° Substitute 5a – 5 for mVTZ. mVTZ =[5(5) – 5)]° = 20° Substitute 5 for a and simplify.

  22. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

  23. Helpful Hint Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms.

  24. Example 3: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

  25. Step 1 Show that EG and FH are congruent. Since EG = FH, Example 3 Continued

  26. Step 2 Show that EG and FH are perpendicular. Since , Example 3 Continued

  27. Step 3 Show that EG and FH are bisect each other. Since EG and FH have the same midpoint, they bisect each other. Example 3 Continued The diagonals are congruent perpendicular bisectors of each other.

  28. Check It Out! Example 4 Given: PQTS is a rhombus with diagonal Prove:

  29. 2. 4. 5. 6. 7. Check It Out! Example 4 Continued 1. PQTSis a rhombus. 1. Given. 2. Rhombus → each diag. bisects opp. s 3. QPR  SPR 3. Def. of  bisector. 4. Def. of rhombus. 5. Reflex. Prop. of  6. SAS 7. CPCTC

  30. Lesson Quiz: Part I A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length. 1.TR2.CE 35 ft 29 ft

  31. Lesson Quiz: Part II PQRS is a rhombus. Find each measure. 3.QP4. mQRP 42 51°

  32. Class work problems Page 402: 10, 12,14,16,18,20 Page 412 :10, 14, 16, 17, 22, 24

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