A quadrilateral with two opposite sides that are parallel is called a parallelogram.

Opposite sides of a parallelogram are equal AB=CD,AD=BC

The opposite angles of a parallelogram are equal ∠ABC=∠ADC, ∠BAD=∠BCD

Opposite angles of a parallelogram bisect each other AO=CO,BO=DO

A quadrilateral with two equal sides is a parallelogram.

A quadrilateral with two equal opposite angles is a parallelogram

A quadrilateral whose diagonals bisect each other is a parallelogram

A set of quadrilaterals whose opposite sides are parallel and equal is a parallelogram AD//BC, AD=BC

The line segment connecting the midpoints of both sides of a triangle is called the median of the triangle

Triangle Median Theorem

A parallelogram with one right angle is called a rectangle

Angle: The four corners of a rectangle are all right angles ∠ ABC=∠BCD=∠ADC=∠BAC=90°

Sides: Opposite sides are parallel and equal

Diagonals: The diagonals of a rectangle bisect each other and are equal AO=BO=½AC,BO=CO=½BC,AC=BD∴AO=BO=CO=DO

Properties of a right triangle: The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse

There is a parallelogram with right angles

A parallelogram with equal diagonals is a rectangle

A quadrilateral with three right angles is a rectangle

A set of parallelograms with equal adjacent sides is called a rhombus

All four sides of a rhombus are equal AB=BC=CD=AD

The two diagonals of a rhombus bisect each other and are perpendicular to AC⊥BD. AO=CO，BO=DO

Each diagonal bisects a set of diagonals. AC bisects ∠BAD and ∠BCD, and BD bisects ∠ABC and ∠ADC.

A set of parallelograms with equal adjacent sides is a rhombus

A parallelogram with perpendicular diagonals is a rhombus

A quadrilateral with four equal sides is a rhombus

A parallelogram with a set of equal adjacent sides and one right angle is called a square

There is a set of rectangles with equal adjacent sides

A rhombus with one right angle

The four corners of a square are right angles and the four sides are equal

The diagonals of a square are equal and bisect each other perpendicularly

First determine the rhombus and rectangle conditions (choose one of the two)

First determine the rectangular and rhombus conditions (choose one of the two)

Direct determination of parallelograms