Lesson 15 7 Conditions for Rectangles Rhombi and

Lesson 15. 7: Conditions for Rectangles, Rhombi, and Squares

Conditions for Parallelograms 1. If both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram. (definition) 2. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. (Thm) 3. If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. (Thm) 4. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. (Thm)

Definitions: A rectangle is a quadrilateral with 4 right angles. A rhombus is a quadrilateral with 4 congruent sides. A square is a quadrilateral with 4 right angles and 4 congruent sides. Therefore, a square is a rhombus and a rectangle.

Properties of a Rectangle 1. A rectangle is a parallelogram 2. The diagonals of a rectangle are congruent (Thm)

Conditions for a Rectangle 1. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. (Thm) 2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. (Thm)

Properties of a Rhombus 1. A rhombus is a parallelogram 2. The diagonals of a rhombus are perpendicular. (Thm) 3. Each diagonal of a rhombus bisects a pair of opposite angles. (Thm)

Conditions for a Rhombus 1. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. (Thm) 2. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. (Thm) 3. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. (Thm)