1. 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. definitions Rhombus Rectangle Square
A parallelogram with all sides congruent
Rectangle
A parallelogram with all angles congruent
(4 right angles)
Square
and all angles congruent

3. Quadrilaterals
parallelograms
rectangles
rhombi
squares

4. theorem A parallelogram is a rhombus if and only if
its diagonals are perpendicular. B
then it is
a rhombus A C D

5. theorem A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite angles. B
bisects
then it is
a rhombus
and A C
bisects D

6. theorem A parallelogram is a rectangle if and only if
its diagonals are congruent. A B
then it is
a rectangle D C

7. A B D C
Statement Reason

8. Sometimes Example A rectangle is a square.
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
rectangles
rhombi
squares
Sometimes

9. Always Example A square is a rhombus Sometimes, Always, or Never?
Quadrilaterals
parallelograms
rectangles
rhombi
squares
Always

10. Sometimes Example A rhombus is a rectangle
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
rectangles
rhombi
squares
Sometimes

11. Sometimes Example A parallelogram is a rectangle
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
rectangles
rhombi
squares
Sometimes

12. The diagonals of a rectangle bisect each other.
Example
rectangle EFGH, EG = 8z – 16
Find EK and GK E F K H G
The diagonals of a rectangle bisect each other.

13. Example Rectangle ABCD Find the value of x
By definition of a rectangle, D C

14. Example M A H T

15. Example M A H T

16. Example M A H T
perimeter of rect. MATH =

17. 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?