Chapter 6 Quadrilaterals

Section 6.1 Polygons

Polygon A polygon is formed by three or more segments called sides No two sides with a common endpoint are collinear. Each side intersects exactly two other sides, one at each endpoint. Each endpoint of a side is a vertex of the polygon. Polygons are named by listing the vertices consecutively.

Identifying polygons State whether the figure is a polygon. If not, explain why.

Polygons are classified by the number of sides they have TYPE OF POLYGON 3 4 5 6 7 NUMBER OF SIDES TYPE OF POLYGON 8 9 10 12 N-gon octagon triangle nonagon quadrilateral pentagon decagon dodecagon hexagon heptagon N-gon

Two Types of Polygons: Convex: If a line was extended from the sides of a polygon, it will NOT go through the interior of the polygon. Example:

2. Concave: If a line was extended from the sides of a polygon, it WILL go through the interior of the polygon. Example:

Regular Polygon A polygon is regular if it is equilateral and equiangular A polygon is equilateral if all of its sides are congruent A polygon is equiangular if all of its interior angles are congruent

Diagonal A segment that joins two nonconsecutive vertices.

Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior angles of a quadrilateral is 360° 1 4 2 3

Properties of Parallelograms Section 6.2 Properties of Parallelograms

Parallelogram A quadrilateral with both pairs of opposite sides parallel

Theorem 6.2 Opposite sides of a parallelogram are congruent.

Theorem 6.3 Opposite angles of a parallelogram are congruent

Theorem 6.4 Consecutive angles of a parallelogram are supplementary. 2 1 2 3 4

Theorem 6.5 Diagonals of a parallelogram bisect each other.

Proving Quadrilaterals are Parallelograms Section 6.3 Proving Quadrilaterals are Parallelograms

Theorem 6.6 To prove a quadrilateral is a parallelogram: Both pairs of opposite sides are congruent

Theorem 6.7 To prove a quadrilateral is a parallelogram: Both pairs of opposite angles are congruent.

Theorem 6.8 To prove a quadrilateral is a parallelogram: An angle is supplementary to both of its consecutive angles. 1 2 3 4

Theorem 6.9 To prove a quadrilateral is a parallelogram: Diagonals bisect each other.

Theorem 6.10 To prove a quadrilateral is a parallelogram: One pair of opposite sides are congruent and parallel. > >

Types of parallelograms Section 6.4 Types of parallelograms

Rhombus Parallelogram with four congruent sides.

Properties of a rhombus Diagonals of a rhombus are perpendicular.

Properties of a rhombus Each Diagonal of a rhombus bisects a pair of opposite angles.

Rectangle Parallelogram with four right angles.

Properties of a rectangle Diagonals of a rectangle are congruent.

Square Parallelogram with four congruent sides and four congruent angles. Both a rhombus and rectangle.

Properties of a square Diagonals of a square are perpendicular.

Properties of a square Each diagonal of a square bisects a pair of opposite angles. 45° 45° 45° 45° 45° 45° 45° 45°

Properties of a square Diagonals of a square are congruent.

3-Way Tie Rectangle Rhombus Square

Section 6.5 Trapezoids and Kites

Trapezoid Quadrilateral with exactly one pair of parallel sides. Parallel sides are the bases. Two pairs of base angles. Nonparallel sides are the legs. Base > Leg Leg > Base

Isosceles Trapezoid Legs of a trapezoid are congruent.

Theorem 6.14 > Base angles of an isosceles trapezoid are congruent.

Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. > A B > D C ABCD is an isosceles trapezoid

Theorem 6.16 Diagonals of an isosceles trapezoid are congruent. > > A B C D ABCD is isosceles if and only if

Examples on Board

Midsegment of a trapezoid Segment that connects the midpoints of its legs. Midsegment

Midsegment Theorem for trapezoids Midsegment is parallel to each base and its length is one half the sum of the lengths of the bases. A B C D M N MN= (AD+BC)

Examples on Board

Kite Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorem 6.18 Diagonals of a kite are perpendicular. A B C D

Theorem 6.19 In a kite, exactly one pair of opposite angles are congruent. A B C D

Examples on Board

Pythagorean Theorem c a b

Special Quadrilaterals Section 6.6 Special Quadrilaterals

Properties of Quadrilaterals Property Rectangle Rhombus Square Trapezoid Kite Both pairs of opposite sides are congruent Diagonals are congruent Diagonals are perpendicular Diagonals bisect one another Consecutive angles are supplementary Both pairs of opposite angles are congruent X X X X X X X X X X X X X X X X X X X X X

Properties of Quadrilaterals Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition? PARALLELOGRAM RECTANGLE ISOSCELES TRAPEZOID SQUARE RHOMBUS

Areas of Triangles and Quadrilaterals Section 6.7 Areas of Triangles and Quadrilaterals

Area Congruence Postulate If two polygons are congruent, then they have the same area.

Area Addition Postulate The area of a region is the sum of the areas of its non-overlapping parts.

Area Formulas TRIANGLE RECTANGLE SQUARE PARALLELOGRAM A=bh A=lw

Area Formulas RHOMBUS KITE

Area Formulas TRAPEZOID h