Honors Geometry Unit 8 Quadrilaterals

Consecutive vertices

two vertices that are connected by sides of a polygon

Diagonal segments

Segments that connect non-consecutive vertices

Regular polygons

equilateral and equiangular polygons

Polygon interior angles theorem

The sum of the interior angles of a polygon is (the number of sides minus 2) times 180; 180 x (n-2)

Polygon exterior angles theorem

The sum of the exterior angles (one at each vertex) is always 360 degrees

Parallelogram (definition/necessity)

If a quadrilateral is a parallelogram, then its opposite sides are parallel

Necessities of a parallelogram

1. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. 2. If a quadrilateral is a parallelogram, then its opposite sides are congruent. 3. If a quadrilateral is a parallelogram, then its opposite angles are congruent. 4. If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Sufficiencies of a parallelogram

1. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 2. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 3. If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. 4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 5. One angle is supplementary to both adjacent angles (not in notes).

Rhombus

A quadrilateral with 4 congruent sides

Rectangle

A quadrilateral with 4 congruent angles

Square

A quadrilateral with 4 congruent sides and 4 congruent angles

Necessities for rhombi

If a polygon is a rhombus, then it has 4 congruent sides, perpendicular diagonals, diagonals that bisect the interior angles, and diagonals bisect each other (not in notes)

Sufficiencies for rhombi

If a quadrilateral has 4 congruent sides or perpendicular diagonals, then it is a rhombus

Necessities for rectangles

If a polygon is a rectangle, then it has 4 congruent angles (BONUS: all angles are right) and congruent diagonals

Sufficiencies for rectangles

If a quadrilateral has 4 congruent angles (BONUS: 4 right angles) or congruent diagonals, then it is a rectangle

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

Bases: The 2 parallel sides

Legs: Non parallel sides

Base angles: 2 angles on the same base

Midsegment of a trapezoid

The segment that connects the midpoints of a trapezoid's legs

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 bases

Isosceles trapezoid

A trapezoid with congruent legs

Necessities of and isosceles trapezoid

The legs are congruent, the base angles are congruent, and the diagonals are congruent

Sufficiencies of an isosceles trapezoid

If a trapezoid has congruent legs, congruent bases, or congruent diagonals, then it is isosceles

Kite

A quadrilateral with 2 pairs of congruent consecutive sides, but none of the opposite sides are congruent

Necessities of a kite

If a polygon is a kite, then the diagonals are perpendicular, it has exactly one pair of congruent angles, and one diagonal bisects the other