Chapter 7 Postulates and Theorems

Theorem 7.1 (Polygon Interior Angles Sum Theorem)

The sum of the interior angle measures of an n-sided convex polygon is (n-2) x 180°.

Theorem 7.2 (Polygon Exterior Angles Sum Theorem)

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

Theorem 7.3

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

Theorem 7.4

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

Theorem 7.5

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

Theorem 7.6

If a parallelogram has one right angle, then it has four right angles.

Theorem 7.7

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

Theorem 7.8

If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles.

Theorem 7.9

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

Theorem 7.10

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

Theorem 7.11

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

Theorem 7.12

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

Theorem 7.13

If a parallelogram is a rectangle, then its diagonals are congruent.

Theorem 7.14

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Theorem 7.15

If a parallelogram is a rhombus, then its diagonals are perpendicular.

Theorem 7.16

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Theorem 7.17

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorem 7.18

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Theorem 7.19

If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Theorem 7.20

If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Theorem 7.21

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

Theorem 7.22

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

Theorem 7.23

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

Theorem 7.24 (Trapezoid Midsegment Theorem)

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

Theorem 7.25

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

Theorem 7.26

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