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

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

Quadrilaterals parallelograms rectangles rhombi squares

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

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

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

A B D C Statement Reason

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

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

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

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

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.

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

Example M A H T

Example M A H T

Example M A H T perimeter of rect. MATH =

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?