Polygons and Quadrilaterals Chapter 6 Packet Polygons and Quadrilaterals

Sketch a convex pentagon, hexagon and heptagon Sketch a convex pentagon, hexagon and heptagon. For each figure, draw all the diagonals you can from one vertex. What conjecture can you make about the relationship between the number of sides of a polygon and the number of triangles formed by the diagonals from one vertex?

6.1 The Polygon Angle-Sum Theorems Theorem 6-1 Polygon Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is: ___(n - 2)180____.

Example 1: Finding a Polygon Angle Sum 1) What is the sum of the interior angle measures of a 17-gon? 2) What is the sum of the interior angle measures of a nonagon? 3) The sum of the interior angle measures of a polygon is 1980. How many sides does the polygon have?

4) What is the measure of angle G in quadrilateral EFGH?

Equilateral Equiangular Equal sides Equal angles

Corollary to the Polygon Angle-Sum Theorem (How to find 1 Interior Angle) The measure of each interior angle of a regular n-gon is: (n-2)(180) n

Example 2: Using the Polygon Angle-Sum Theorem to Find an Interior Angle 1) Find the measure of one interior angle of a regular heptagon. 2) Find the measure of one interior angle of a regular 360-gon.

Theorem 6-2: Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, on at each vertex, is 360. For the pentagon,

Example 3: Finding an Exterior Angle Measure 1) Find the measure of an exterior angle of a regular pentagon. 2) Find the measure of an exterior angle of a regular 72-gon.

Example 3: Finding an Exterior Angle Measure (part 2) 3) The measure of an exterior angle of a regular polygon is 12. a. What is the measure of an interior angle? b. How many sides does the polygon have?

What do you remember?

Ch 6.2 Properties of Parallelograms Opposite Sides / Angles: Consecutive Angles:

Ch 6.2 Properties of Parallelograms Opposite Sides / Angles: Consecutive Angles: are congruent  are supplementary (add up to 180)

Example 1: Using Consecutive Angles Draw a parallelogram and label it PQRS. If the measure of angle S is 86, then what is the measure of angle R? Step 1: Consecutive Angles are _________________________, so __________________________________.   Step 2: Substitute Step 3: Solve to find angle R.

Example 2: Using Algebra to Find Lengths What are PR and SQ? Step 1: The diagonals of a parallelogram _bisect_ each other, therefore _PT = TR_ and __ST = TQ__.   Step 2: Set up a system of equations by substituting the algebraic expressions for each segment length.

Continued: Step 3: Use substitution to solve the system of equations. Step 4: Once the system is solved, plug in your answers for your variables to find segment lengths.

6.3 Proving that a Quadrilateral is a Parallelogram

Example 1: Finding Values for Parallelograms Draw a parallelogram and label it ABCD. Label the following segments: AB = 2y + 2, BC = y + 4, CD = 3y – 9, and AD = 3x + 6. For what values of x and y must ABCD be a parallelogram?   Step 1: Opposite sides of a parallelogram are ______, therefore ______=______ and ______=______. Step 2: Set up a system of equations and solve for x and y.  Step 3: Substitute your answers for the variables and make sure the opposite sides are congruent.

Example 1: Finding Values for Parallelograms (continued) 4. For what values of x and y must EFGH be a parallelogram?   Step 1: Opposite angles of a parallelogram are ______, therefore ______=______ and ______=______. Step 2: Set up a system of equations and solve for x and y. Step 3: Substitute your answers for the variables and make sure the opposite sides are congruent.

Chapter 6.4 & 6.5 Properties of & Conditions for Rhombuses, Rectangles, and Squares

Chapter 6.4 & 6.5 Properties of & Conditions for Rhombuses, Rectangles, and Squares sides sides right angles right angles

Example 1: Finding Angle Measures in a Rhombus What are the measure of the numbered angles in rhombus PQRS? Hint: Opposite angles are ____________________, therefore angle S = _______ Hint: Diagonals are __________________, therefore ____________ and _____________ Hint: Triangles = _________ (Write an equation to help you find angles 1, 2, 3, and 4)

Example 2: Finding Diagonal Length in a Rectangle If LN = 4x - 17 and MO = 2x + 13, what are the lengths of the diagonals of rectangle LMNO? Hint: The diagonals of a rectangle are _______, therefore _______=_______. (Write an equation, solve for x, and plug in to find the lengths of the diagonals.)

Example 3: Using Properties of Special Parallelograms 1. For what value of x is ABCD a rhombus? Hint: Diagonals of a rhombus are ______________ Therefore ______________ = _______________   (Write an equation, solve for x, and plug in to find the measures of the angles. Are they congruent?)

Example 3 (continued) 2. For what value of y is DEFG a rectangle? Hint: Diagonals of a rectangle are _______________ Therefore _____________ = ______________ (Write an equation, solve for y, and plug in to find the lengths of the diagonals. Are they congruent?)

Kites!

6.6 Trapezoids and Kites Trapezoid definition: Isosceles Trapezoid:   Midsegment of a Trapezoid: Midsegment is _________________ to its __________________ Length of the midsegment is _________ the ________ of the lengths of its ___________

Example 1: Finding Angle Measures in Trapezoids In the diagram, PQRS is an isosceles trapezoid & the measure of angle R = 106. What are the measures of angles P, Q, and S? Hint: In trapezoids, the side angles are _________________, therefore _________________ Hint: In isosceles trapezoids, the top & bottom angles are _______, therefore ______=_______ & ______=_______ Example 1: Finding Angle Measures in Trapezoids

Example 2: Using the Midsegment of a Trapezoid MN is the midsegment of trapezoid PQRS. What is x? What is MN? Step 1: midsegment = ½ (top + bottom)   Step 2: solve for x Step 3: Plug in x to find segment MN

Example 3: Finding Angle Measures in Kites Quadrilateral KLMN is a kite. What are the measures of angles 1, 2, and 3? Hint: The diagonals in a kite are ____________________, therefore angle 1 = Hint: Angle K is bisected, so angle 3 = Hint: Use triangle = 180 to help you solve for angle 2.

6.7 Polygons in the Coordinate Plane

Example 1: Classifying a Triangle

Example 2: Classifying a Parallelogram

Example 3: Classifying a Quadrilateral An isosceles trapezoid has vertices A(0,0), B(2, 4), C(6,4), and D(8,0). What special quadrilateral is formed by connecting the midpoints of the sides of ABCD? Graph the shape using graph paper Find the midpoint of each side Connect the midpoints What shape is made by connecting the midpoints?