1. created by chris markstrum © 2005 6.4 Rhombi, Rectangles, and Squares California State Standards for Geometry 4: Prove basic theorems involving congruence & similarity. 7: Prove & use theorems involving parallel lines and properties of quadrilaterals. 12: Find & use side and angle measures of triangles and polygons 17: Prove theorems using coordinate geometry

2. created by chris markstrum 2005 definitions Rhombus A parallelogram with all sides congruent Rectangle A parallelogram with all angles congruent (4 right angles) Square A parallelogram with all sides congruent and all angles congruent

3. created by chris markstrum 2005 Quadrilaterals parallelograms rectangles rhombi squaressquares squaressquares

4. created by chris markstrum 2005 theorems about rhombi, rectangles, and squares Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary A quadrilateral is a square if and only if it is both a rectangle and a rhombus.

5. created by chris markstrum 2005 A parallelogram is a rhombus if and only if its diagonals are perpendicular. A B C D then it is a rhombus theorem

6. created by chris markstrum 2005 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. A B C D then it is a rhombus bisects and bisects theorem

7. created by chris markstrum 2005 A parallelogram is a rectangle if and only if its diagonals are congruent. A B C D then it is a rectangle theorem

8. created by chris markstrum 2005 1. Given: Prove: ABCD is a rectangle 1. Given 2. 7. ABCD is a rectangle 7. Def rectangle 3. 4. 4. C.P.C.T.C. 5. 6. 6. Transitive Prop 2. AB C D 5.

9. created by chris markstrum 2005 AB C D StatementReason

10. created by chris markstrum 2005 Quadrilaterals parallelograms rectangles rhombi squaressquares Example A rectangle is a square. Sometimes, Always, or Never? Sometimes

11. created by chris markstrum 2005 Example A square is a rhombus Sometimes, Always, or Never? Always Quadrilaterals parallelograms rectangles rhombi squaressquares

12. created by chris markstrum 2005 Example A rhombus is a rectangle Sometimes, Always, or Never? Quadrilaterals parallelograms rectangles rhombi squaressquares Sometimes

13. created by chris markstrum 2005 Example A parallelogram is a rectangle Sometimes, Always, or Never? Quadrilaterals parallelograms rectangles rhombi squaressquares Sometimes

14. created by chris markstrum 2005 Example rectangle EFGH, EG = 8 z – 16 Find EK and GK E F G H K The diagonals of a rectangle bisect each other.

15. created by chris markstrum 2005 Rectangle ABCD Find the value of x Example By definition of a rectangle, A B C D

16. created by chris markstrum 2005 Example M A T H

17. created by chris markstrum 2005 Example M A T H

18. created by chris markstrum 2005 Example perimeter of rect. MATH = M A T H

19. created by chris markstrum 2005 Summary What are the three ways to show that a parallelogram is a rhombus What are the two ways to show that a parallelogram is a rectangle? How do you show that a parallelogram is a square?