Chapter 6

Formed by 3 or more segments (sides) Each side intersects only 2 other sides (one at each endpoint)

Number of Sides Name of Polygon 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 7Heptagon 8Octagon 9Nonagon 10Decagon 12Dodecagon nn-gon Polygons are named by the number of sides they have

CONVEXCONCAVE

Regular Polygons: Equilateral & Equiangular

Segment that joins 2 non-consecutive vertices.

The Sum of the Measures of the Interior Angles of a Quadrilateral is 360°

Solve for x…

Quadrilateral Both pairs of opposite sides are parallel

OPPOSITE SIDES are congruent If a Quadrilateral is a Parallelogram, Then…. OPPOSITE ANGLES are congruent

CONSECUTIVE ANGLES are supplementary If a Quadrilateral is a Parallelogram, Then…. DIAGONALS bisect each other  A +  B = 180°

If both pairs of opposite sides of a quad. are  … If both pairs of opposite angles of a quad. are  … If an angle of a quad. is supplementary to both of its consecutive angles … If the diagonals of a quad. bisect each other… Then, the Quadrilateral is a Parallelogram. Proving Quadrilaterals are Parallelograms…

If one pair of opposite sides of a quadrilateral are congruent AND parallel Then, the Quadrilateral is a Parallelogram. Proving Quadrilaterals are Parallelograms…

Describe how to prove that ABCD is a parallelogram given that ∆PBQ  ∆RDS and ∆PAS  ∆RCQ. Let’s practice….

Prove that EFGH is a parallelogram by showing that a pair of opposite sides are both congruent and parallel. Use E(1, 2), F(7, 9), G(9, 8), and H(3, 1). Prove that JKLM is a parallelogram by showing that the diagonals bisect each other. Use J(-4, 4), K(-1, 5), L(1, -1), and M(-2, -2).

Sections 1, 2, & 3

A parallelogram with 4 congruent sides Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.

Theorem 6.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC  BD

Theorem 6.12: A parallelogram is a rhombus if and only if its diagonals bisect a pair of opposite angles. ABCD is a rhombus if and only if AD bisects  CAB and  BDC and BC bisects  DCA and  ABD

A parallelogram with 4 right angles Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.

Theorem 6.13: A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if AC  BD

A parallelogram with 4 congruent sides AND 4 right angles Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Quadrilateral with only one pair of parallel sides. Parallel sides are the “bases” Non-parallel sides are the “legs” Has 2 pairs of base angles Base Angles

Show that RSTV is a trapezoid…

Legs are congruent If m  A = 45°, What is the measure of  B? What is the measure of  C? What is the measure of  D?

Theorem 6.14: If a trapezoid is isosceles, then each pair of base angles is congruent  A   D,  B   C

Theorem 6.15: (Converse to theorem 6.14) If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid ABCD is an isosceles trapezoid

Theorem 6.16: A trapezoid is isosceles if and only if its diagonals are congruent ABCD is isosceles if and only if AC  BD

Midsegment Theorem for Trapezoids (Theorem 6.17) EF  AB, EF  DC, EF = ½(AB + DC) The midsegment of a trapezoid is … Parallel to each base ½ the sum of the length of the bases

A quadrilateral that has two pairs of consecutive congruent sides. Opposite sides are NOT congruent.

Theorem 6.18: If a quadrilateral is a kite, then its diagonals are perpendicular KT  EI

If KS = ST = 5, ES = 4, and KI = 9, What is the measure of EK? What is the measure of SI?

Theorem 6.19: If a quadrilateral is a kite, then only one pair of opposite angles are congruent  K  M,  J   L

If m  J = 70 and m  L = 50, What is m  M & m  K?

Sections 4 & 5

When you join the midpoints of the sides of ANY quadrilateral, what special quadrilateral is formed? Explain. On a piece of graph paper… Draw ANY quadrilateral Find and connect the midpoints of each side What type of Quadrilateral is formed? How do you know?

Let’s prove a quadrilateral is a “special” shape… Use the Definition of the Shape Use a Theorem EXAMPE: Show that PQRS is a rhombus How would you prove this to be true?

Create a Graphic Organizer showing the relationship between the following figures… Isosceles Trapezoid Kite Parallelogram Quadrilaterals Rectangle Rhombus Square Trapezoid Requirements.. Accurate Graphic Organizers Each figure should include an picture and description Bold, Clear, and Colorful

Graphic Organizer Examples

A = bh Find the area of the rectangle below:

A = bh Find the area of the parallelogram below:

A = ½bh Find the area of the triangles below:

What is the height the triangle below: A = 27 ft 2 Base = 9 feet

A = ½h(b 1 + b 2 ) Find the area of the trapezoid below:

A = ½d 1 d 2 Find the area of the kite below:

A = ½d 1 d 2 Find the area of the rhombus below:

Sections 6 & 7