Y. Davis Geometry Notes Chapter 6

Diagonal A segment that connects 2 non-consecutive vertices of a polygon

Theorem 6.1 Polygon Interior Angles Sum The sum of the interior angles of any convex n-gon is

Theorem 6.2 Polygon Exterior Angles Sum The sum of the exterior angle measures a convex polygon, one angle at each vertex, is 360 degrees.

Parallelogram A quadrilateral with both pair of opposite sides parallel.

Properties of parallelograms Theorem 6.3—If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem 6.4—If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem 6.5—If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem 6.6—If a parallelogram has 1 right angle, then it has 4 right angles.

Diagonals of Parallelogram Theorem 6.7—If a quadrilateral is a parallelogram, then its diagonals bisect each other. Theorem 6.8—If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into 2 congruent triangles.

Conditions for Parallelograms Theorem 6.9—If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. Theorem 6.10—If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. Theorem 6.11—If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Theorem 6.12—If one pair of opposite sides of a quadrilateral is both parallel and congruent, then it is a parallelogram.

Rectangle A parallelogram with 4 right angles. Opposite sides parallel and congruent Opposite angles congruent Consecutive angles are supplementary Diagonals bisect each other.

Theorem 6.13 diagonals of a rectangle If a parallelogram is a rectangle, then its diagonals are congruent.

Theorem 6.14 Prove parallelograms are rectangles If the diagonals of a parallelogram are congruent, then it is a rectangle.

Rhombus A parallelogram with 4 congruent sides Opposite sides congruent and parallel Opposite angles congruent. Consecutive sides supplementary Diagonals bisect each other.

Diagonals of a Rhombus Theorem 6.15—If a parallelogram is a rhombus, then its diagonals are perpendicular. Theorem 6.16—If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Square A parallelogram with 4 congruent sides and 4 right angles. Opposite sides parallel and congruent. Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other. Diagonals congruent Diagonals perpendicular Diagonals bisect each pair of opposite angles.

Conditions for Rhombi Theorem 6.17—If the diagonals of a parallelogram are perpendicular, then it is a rhombus. Theorem 6.18—If a diagonal of a parallelogram bisect a pair of opposite angles, then it is a rhombus. Theorem 6.19—If one pair of consecutive sides of a parallelogram are congruent, then it is a rhombus

Conditions for Squares Theorem 6.20—If a parallelogram is both a rectangle and rhombus, then it is a square.

Trapezoids A quadrilateral with exactly one pair of opposite sides parallel. Bases—parallel sides Legs—non-parallel sides 2 pair of Base Angles–each pair includes a base.

Isosceles Trapezoid A trapezoid with congruent legs. Theorem 6.21—If a trapezoid is isosceles, then each pair of base angels are congruent. Theorem 6.22—If a trapezoid has one pair of congruent base angles, then it is isosceles. Theorem 6.23—A trapezoid is isosceles if and only if its diagonals are congruent.

Midsegment of aTrapezoid The segment that connects the midpoints of the legs of a trapezoid.

Theorem 6.24 Trapezoid Midsegment Theorem The Midsegment of a trapezoid is parallel to the 2 bases and ½ the sum of the bases.

Kite A quadrilateral with exactly 2 pairs of consecutive sides congruent, but no opposite sides congruent.

Kites Theorem 6.25—If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 6.26—If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.