Geometry: Polygons and Parallelograms

Polygons

• A polygon is a closed figure consisting of straight segments.
• A diagonal is a segment that joins nonconsecutive vertices of a polygon.
• The sum of the interior angles of a polygon with n sides is (n-2)180°.
• The sum of the angles of a triangle is 180°.
• An equilateral polygon has all sides congruent.
• An equiangular polygon has all angles congruent.
• A regular polygon has all sides and all angles congruent.
• The sum of the measures of exterior angles of a convex polygon is 360°.

Parallelograms

• A parallelogram is a quadrilateral with opposite sides parallel.
• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.
• Consecutive angles in a parallelogram are supplementary.
• The diagonals of a parallelogram bisect each other.

Proving Parallelograms

• If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
• If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Special Parallelograms

• A rhombus is a parallelogram with four congruent sides.
• A rectangle is a parallelogram with four right angles.
• A square is both a rhombus and a rectangle.
• The diagonals of a rhombus are perpendicular bisectors to each other.
• The diagonals of a rhombus bisect the opposite angles.
• The diagonals of a rectangle are congruent.

Trapezoids and Kites

• A trapezoid is a quadrilateral with exactly one pair of parallel sides.
• The parallel sides in a trapezoid are called the bases.
• The non-parallel sides in a trapezoid are called the legs.
• An isosceles trapezoid is a trapezoid with congruent legs.
• Base angles of an isosceles trapezoid are congruent.
• The diagonals of an isosceles trapezoid are congruent.
• The midsegment of a trapezoid connects the midpoints of each leg. It is parallel to the bases and its length is the average of the lengths of the bases.
• A kite is a quadrilateral with two pairs of consecutive congruent sides.
• The diagonals of a kite are perpendicular.
• One pair of opposite angles in a kite is congruent.

Hierarchy of Quadrilaterals

• All squares are rectangles.
• All rectangles are parallelograms.
• All rhombuses are parallelograms.
• All parallelograms are quadrilaterals.
• All kites are quadrilaterals.
• All trapezoids are quadrilaterals.