Theorems & Corollaries

• Polygon Interior Angles Theorem: the sum of the measures of the interior angles of a convex n-gon is (n-2) * 180

• Polygon Exterior Angles Theorem: the sum of the measures of the exterior angles of a convex polygon (one at each vertex), is 360^o

• If a quadrilateral is a parallelogram, then its opposite sides are congruent

• If a quadrilateral is a parallelogram, then its opposite angles are congruent

• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

• If a quadrilateral is a parallelogram, then its diagonals bisect each other

• Rhombus Corollary: a quadrilateral is a rhombus if and only if it has 4 congruent sides

• Rectangle Corollary: a quadrilateral is a rectangle if and only if it has 4 right angles

• Square Corollary: a quadrilateral is a square if and only if it is a rhombus and a rectangle (4 congruent sides & 4 right angles)

• A parallelogram is a rhombus if and only if its diagonals are perpendicular

• A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles

• A parallelogram is a rectangle if and only if its diagonals are congruent

• If a trapezoid is isosceles, then each pair of base angles are congruent

• If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid

• A trapezoid is isosceles if and only if its diagonals are congruent

• Midsegment Theorem for Trapezoids: the midsegment of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the base

  • Midsegment = 1/2(b1 + b2)

• If a quadrilateral is a kite, then its diagonals are perpendicular

• If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent

  • The angles that connect the non-congruent sides are congruent

Properties of Parallelograms

• Opposite angles are congruent

• Consecutive angles are supplementary

• Opposite sides are parallel

• Opposite sides are congruent

• Diagonals bisect each other

Properties of Rhombuses

• All of the parallelogram properties plus

• All sides are congruent

• Diagonals bisect a pair of opposite angles

• Diagonals are perpendicular

Properties of Rectangles

• All of the parallelogram properties plus

• All angles are congruent

• All angles equal 90^o

Properties of Squares

• Has the properties of rectangles and rhombuses

Definitions

• Consecutive vertices: two vertices that are endpoints of the same side of a polygon

• Diagonal: a segment that joins two nonconsecutive vertices of a polygon

• Interior Angles of a Quadrilateral: the sum of the measures of the interior angles of a quadrilateral is 360^o

• Parallelogram: a quadrilateral with both pairs of opposite sides parallel

• Rhombus: a parallelogram with 4 congruent sides

• Rectangle: a parallelogram with 4 right angles

• Square: a parallelogram with 4 congruent sides and 4 right angles

• Trapezoid: a quadrilateral with exactly one pair of parallel sides

• Bases: the parallel sides of a trapezoid

• Legs: the nonparallel sides of a trapezoid

• Isosceles trapezoid: the legs are congruent

• Midsegment of a trapezoid: the segment that connects the midpoints of its legs

• Kite: a quadrilateral that has two pairs of consecutive congruent sides, but opposite are not congruent