6.3/4 Rhombuses, Rectangles, and Squares

Three Definitions 1.A rhombus is a parallelogram with four congruent sides. 1.A rectangle is a parallelogram with four right angles. 1.A square is a parallelogram with four congruent sides and four right angles.

Venn Diagram for Parallelograms Label each figure with one of the following terms so that the diagram represents the relationships between the different types of polygons: quadrilaterals, parallelograms, rectangles, rhombuses, and squares. Quadrilaterals Rhombuses Rectangles SQUARESSQUARES Parallelograms

Two Properties of a Rhombus 1. The diagonals of a rhombus are perpendicular. 2. The diagonals of a rhombus bisect the angles. A quadrilateral is a rhombus if and only if it has four___________________ sides. (based on definition) A parallelogram is a rhombus if and only if its diagonals are_________________. A parallelogram is a rhombus if and only if each diagonal ________________ a pair of opposite angles. congruent perpendicular bisects

One Property of a Rectangle The diagonals of a rectangle are congruent. A quadrilateral is a rectangle if and only if it has four _____________ angles. (based on definition) A parallelogram is a rectangle if and only if its diagonals are _______________. congruent right

Squares A square is a parallelogram with four congruent sides and four right angles. A square is both a rhombus and a rectangle; therefore, it has all the properties of a rhombus and all the properties of a rectangle. A quadrilateral is a square if and only if it is both a ___________________ and a __________________________. rhombus rectangle

Summary of Properties Since rhombuses, rectangles, and squares are parallelograms, then: opposite sides are parallel. opposite sides are congruent. opposite angles are congruent. consecutive angles are supplementary. diagonals bisect each other. Other properties: Rhombuses All sides congruent Diagonals are perpendicular. Diagonals bisect the angles. Rectangles All right angles. Diagonals are congruent. Squares All sides congruent All right angles. Diagonals are perpendicular. Diagonals bisect the angles. Diagonals are congruent.

Example Solution

More Examples Fill in each statement with always, sometimes, or never. 1. A rhombus is ________________ a square. 2. A rectangle is _______________ a parallelogram. 3. A quadrilateral is __________________ a rhombus. 4. A square is ___________________ a rectangle. sometimes always sometimes always