1. 6.5 Rhombi and Squares

2. Then/Now 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.

3. Vocabulary Rhombus: a parallelogram with all four sides congruent

4. Vocabulary Square: a parallelogram with four congruent sides and four right angles

5. Vocabulary Theorem 6.15: Diagonals of a Rhombus #1 If a parallelogram is a rhombus, then its diagonals are perpendicular.

6. Vocabulary Theorem 6.16: Diagonals of a Rhombus #2 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

7. Vocabulary Theorem 6.17: Condition #1 for a Rhombus If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. *Converse of 6.15

8. Vocabulary Theorem 6.18: Condition #2 for a Rhombus If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. *Converse of 6.16

9. Vocabulary Theorem 6.19: Condition #3 for a Rhombus If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

10. Vocabulary Theorem 6.20: Square Conditions If a quadrilateral is both a rectangle and a rhombus, then it is a square.

11. Example 1A Use Properties of a Rhombus A. The diagonals of rhombus WXYZ intersect at V. If m  WZX = 39.5, find m  ZYX. Answer: m  ZYX = 101

12. Example 1B Use Properties of a Rhombus B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x. Answer: x = 4

13. Example 1A A.m  CDB = 126 B.m  CDB = 63 C.m  CDB = 54 D.m  CDB = 27 A. ABCD is a rhombus. Find m  CDB if m  ABC = 126.

14. Example 1B A.x = 1 B.x = 3 C.x = 4 D.x = 6 B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

15. Rectangle Rhombi

17. Example 2 Is there enough information given to prove that ABCD is a rhombus? Given:ABCD is a parallelogram. AD  DC Prove:ABCD is a rhombus Proofs Using Properties of Rhombi and Squares A.Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. B.No, you need more information

18. Example 3 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. 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?