Geometry - Chapter 7 Theorems

Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n-gon is (n-2) ⋅ 180°.

Corollary to the Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a quadrilateral is 360°.

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°.

Parallelogram Opposite Sides Theorem

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

Parallelogram Opposite Angles Theorem

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

Parallelogram Consecutive Angles Theorem

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

Parallelogram Diagonals Theorem

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

Parallelogram Opposite Sides Converse

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Parallelogram Opposite Angles Converse

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Opposite Sides Parallel and Congruent Theorem

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

Parallelogram Diagonals Converse

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

rhombus Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

rectangle Corollary

A quadrilateral is a rectangle if and only if it has four right angles.

square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Rhombus Diagonals Theorem

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

Rhombus Opposite Angles Theorem

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

Rectangle Diagonals Theorem

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

Isosceles Trapezoid Base Angles Theorem

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

Isosceles Trapezoid Base Angles Converse

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

Isosceles Trapezoid Diagonals Theorem

A trapezoid is isosceles if an only if its diagonals are congruent.

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to each base, and its length is one-half the sum of the lengths of the bases.

kite Diagonals Theorem

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

kite Opposite Angles Theorem

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