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6-5 Rhombi and Squares

6-5 Rhombi and Squares. You determined whether quadrilaterals were parallelograms and/or rectangles. Recognize and apply the properties of rhombi and squares. Determine whether quadrilaterals are rectangles, rhombi, or squares. Properties of a rhombus.

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6-5 Rhombi and Squares

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  1. 6-5 Rhombi and Squares You determined whether quadrilaterals were parallelograms and/or rectangles. • Recognize and apply the properties of rhombi and squares. • Determine whether quadrilaterals are rectangles, rhombi, or squares.

  2. Properties of a rhombus • A rhombus is a parallelogram with all four sides congruent. • A rhombus has all the properties of a parallelogram: • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • A pair of opposite sides are both parallel and congruent. • AND…..

  3. Rhombus • Draw a rhombus on your paper. • Draw diagonals in your rhombus. • Measure the diagonals. Are the diagonals congruent? • Are the diagonals perpendicular? A parallelogram is a rhombus if and only if its diagonals are perpendicular.

  4. Page 430

  5. Since WXYZ is a rhombus, diagonal ZX bisects WZY. Therefore, mWZY = 2mWZX. So, mWZY = 2(39.5) or 79.Since WXYZ is a rhombus, WZ║XY, and ZY is a transversal. A. The diagonals of rhombus WXYZ intersect at V.If mWZX = 39.5, find mZYX. mWZY + mZYX = 180 Consecutive Interior Angles Theorem 79 + mZYX = 180 Substitution mZYX = 101 Subtract 79 from both sides. Answer:mZYX = 101

  6. WX WZ By definition, all sides of a rhombus are congruent. B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x. 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

  7. A. ABCD is a rhombus. Find mCDB if mABC = 126. A.mCDB = 126 B.mCDB = 63 C.mCDB = 54 D.mCDB = 27

  8. Properties of Squares A square is a parallelogram with four congruent sides and four right angles.

  9. Square • Draw a square on your paper. • Draw diagonals in your square. • Measure the diagonals. Are the diagonals congruent? • Are the diagonals perpendicular? A parallelogram is a square if and only if its diagonals are perpendicular and congruent.

  10. This venn diagram summarizes the relationships among parallelograms, rhombi, rectangles, and squares. Page 431

  11. Page 432

  12. Proof: Since it is given that LMNP is a parallelogram, LM║PN and 1 and 5 are alternate interior angles. Therefore, 1  5. It is also given that 1  2 and 2  6, so 1  6 by substitution and 5  6 by substitution. Answer: Therefore, LN bisects L and N. By Theorem 6.18, LMNP is a rhombus. Write a paragraph proof. Given: LMNP is a parallelogram.1  2 and 2  6 Prove:LMNP is a rhombus.

  13. Given: ABCD is a parallelogram.AD DC Prove:ADCD is a rhombus Is there enough information given to prove that ABCD is a rhombus? A. Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. B. No, you need more information.

  14. Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square? A. The diagonal bisects a pair of opposite angles. B. The diagonals bisect each other. C. The diagonals are perpendicular. D. The diagonals are congruent.

  15. Recap… Rhombi: 4 sides Opp sides Opp sides Opp Diag bisect other Diag Diag bisect opp Squares: 4 sides 4 Opp sides Opp sides Opp Diag bisect other Diag Rectangles: 4 rt Opp sides Opp sides Opp Diag bisect other Diag Parallelograms: Oppsides Opp sides Opp Diag bisect other

  16. 6.5 Assignment Page 435, 7-12, 23-33

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