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Rhombuses, Rectangles, and Squares

Rhombuses, Rectangles, and Squares . Section 6.4. Objectives. Use properties of special types of parallelograms. Key Vocabulary. Rhombus Rectangle Square . Theorems. Corollaries Rhombus Corollary Rectangle Corollary Square Corollary 6.10 Diagonals of a Rhombus

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Rhombuses, Rectangles, and Squares

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  1. Rhombuses, Rectangles, and Squares Section 6.4

  2. Objectives • Use properties of special types of parallelograms.

  3. Key Vocabulary • Rhombus • Rectangle • Square

  4. Theorems • Corollaries • Rhombus Corollary • Rectangle Corollary • Square Corollary • 6.10 Diagonals of a Rhombus • 6.13 Diagonals of a Rectangle

  5. Special Parallelograms • In this section we will study 3 special types of parallelograms. • Rhombus • Rectangle • Square

  6. Rhombuses Or Rhombi What makes a quadrilateral a rhombus?

  7. Rhombuses Or Rhombi A rhombusis an equilateral parallelogram. • Four congruent sides.

  8. Rhombus Corollary • If a quadrilateral has four congruent sides, then it is a rhombus. A B D C

  9. Rhombi • Since rhombi are parallelograms, they have all the properties of a parallelogram. • Opposite sides are || and ≅. • Opposite s are ≅. • Consecutive s are supplementary. • Diagonals bisect each other. • In addition, rhombi have one other property, which is a theorem.

  10. Theorem 6.10 • The diagonals of a rhombus are perpendicular. • If ABCD is a rhombus then AC BD. B C A D

  11. m m m m m m m m A A B C C D D B 6. Consecutive angles are supplementary. + + + + = = = = 180° 180° 180° 180° Properties of a RHOMBUS A 1. Two pairs of parallel sides. 2. All sides are congruent. 3. Diagonals are NOT congruent B D 4. Diagonals bisect each other 5. Diagonals form a right angle C

  12. Example 1 The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

  13. Example 1 WX WZ By definition, all sides of a rhombus are congruent. WX = WZ Definition of congruence 8x – 5= 6x + 3 Substitution 2x – 5 = 3 Subtract 6x from each side. 2x = 8 Add 5 to each side. x = 4 Divide each side by 4. Answer:x = 4

  14. Your Turn: ABCDis a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x. A.x = 1 B.x = 3 C.x = 4 D.x = 6

  15. Use rhombus LMNP to find the value of y if N Example 2

  16. N Example 2 The diagonals of a rhombus are perpendicular. Substitution Add 54 to each side. Take the square root of each side. Answer:The value of y can be 12 or –12.

  17. Your Turn: Use rhombus ABCD and the given information to find the value of each variable. Answer: 8 or –8

  18. Rectangles What makes a quadrilateral a rectangle?

  19. Rectangles A rectangleis an equiangular parallelogram. • All angles are congruent. • What must each angle be then? • Four right angles.

  20. Rectangle Corollary • If a quadrilateral has four right angles, then it is a rectangle.

  21. Rectangles • Since rectangles are parallelograms, they have all their properties: • Opposite sides are || and ≅. • Opposite s are ≅. • Consecutive s are supplementary. • Diagonals bisect each other. • In addition, rectangles have their own special property, which leads us to our next theorem.

  22. Theorem 6.11 Diagonals of a Rectangle are congruent.

  23. Review: Properties of Rectangles • All four s are right s(def. of rectangle). • Opposite sides are || and ≅ (prop. of parallelogram). • Opposite s are ≅ (prop. of parallelogram). • Consecutive s are supplementary (prop. of parallelogram). • Diagonals bisect each other (prop. of parallelogram). • Diagonals are ≅ (theorem 6.11).

  24. Quadrilateral RSTU is a rectangle. If and find x. Example 3:

  25. The diagonals of a rectangle are congruent, Example 3: Definition of congruent segments Substitution Subtract 6x from each side. Add 4 to each side. Answer: 8

  26. Quadrilateral EFGH is a rectangle. If and find x. Your Turn: Answer: 5

  27. Example 4 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.

  28. Example 4 Since JKLM is a rectangle, it is a parallelogram. The diagonals of a parallelogram bisect each other, so LN = JN. LN = 6.5 feet JN + LN = JL Segment Addition LN + LN = JL Substitution 2LN = JL Simplify. 2(6.5) = JL Substitution 13 = JL Simplify.

  29. JLKM If a is a rectangle, diagonals are . Example 4 JL=KM Definition of congruence 13 =KM Substitution Answer:KM = 13 feet

  30. Your Turn: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. A. 3 feet B. 7.5 feet C. 9 feet D. 12 feet

  31. Example 5 Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

  32. Since RSTU is a rectangle, it has four right angles. So, mTUR = 90. The diagonals of a rectangle bisect each other and are congruent, so PT  PU. Since triangle PTU is isosceles, the base angles are congruent so RTU  SUT and mRTU = mSUT. Example 5 mRTU = 8x + 4 mSUR = 3x – 2 mSUT + mSUR= 90 Angle Addition mRTU + mSUR= 90 Substitution 8x + 4 + 3x – 2 = 90 Substitution 11x + 2 = 90 Add like terms.

  33. Example 5 11x = 88 Subtract 2 from each side. x = 8 Divide each side by 11. Answer:x = 8

  34. Your Turn: Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x. A.x = 1 B.x = 3 C.x = 5 D.x = 10

  35. Example 6a: Quadrilateral LMNP is a rectangle. Findx.

  36. Example 6a: Angle Addition Theorem Substitution Simplify. Subtract 10 from each side. Divide each side by 8. Answer: 10

  37. Example 6b: Quadrilateral LMNP is a rectangle. Findy.

  38. Example 6b: Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent. Alternate Interior Angles Theorem Substitution Simplify. Subtract 2 from each side. Divide each side by 6. Answer: 5

  39. Your Turn: Quadrilateral EFGH is a rectangle. a. Findx. b. Find y. Answer: 11 Answer: 7

  40. Squares What makes a quadrilateral a square?

  41. Definition: Square • A square is a parallelogram with four congruent sides and four right angles.

  42. Squares A squareis a regular parallelogram. • All angles are congruent • All sides are congruent

  43. Square Corollary If a quadrilateral has four congruent sides and four right angles, then it is a square.

  44. Venn Diagram Shows the relationships between some members of the parallelogram family.

  45. m m m m m m m m B B C D D A A C 8. Consecutive angles are supplementary. + + + + = = = = 180° 180° 180° 180° Properties of a SQUARE 1. Two pairs of parallel sides. A B 2. All sides are congruent. 3. All angles are right. 4. Diagonals are congruent 5. Diagonals bisect each other D C 6. Diagonals form a right angle 7. Opposite angles are congruent.

  46. 2 2 5X+5 EX. 7 D G DEFG is a square DG = 5X + 5 EF = 7X – 19 Find the value for X and the lenght of the side. E F 7X – 19 Since all sides are congruent: Now since all sides are congruent, we need to find the length of just one side: 5X + 5 = 7X – 19 -5X -5X DG = 5X + 5 5 = 2X – 19 = 5( ) + 5 12 +19 +19 = 60 + 5 24 = 2X = 65 X=12 The length of the side is 65.

  47. 3 3 Your Turn: 9X – 3 K H HIJK is a square KH =9X – 3 IJ = 6X + 24 Find the value for X and the lenght of the side. J I 6X + 24 Since all sides are congruent: Now since all sides are congruent, we need to find the length of just one side: 9X – 3 = 6X + 24 -6X -6X KH = 9X – 3 3X – 3 = 24 = 9( ) – 3 +3 +3 9 = 81 – 3 3x = 27 = 78 X=9 The length of the side is 78.

  48. Summary of Properties Parallelogram, Rectangle, Rhombus, and Square

  49. Quadrilateral Relationships 1. Opposite sides parallel. 2. Opposite sides congruent. 3. Opposite angles are congruent. 4. Consecutive ∠s are supplementary. 5. Diagonals bisect each other. 1. Has 4 Congruent sides. 2. Diagonals are perpendicular. 1. Has 4 right angles. 2. Diagonals are congruent. 1. 4 congruent sides 2. 4 congruent (right) ∠s

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