Ch. 6 Notes6.1: Polygon Angle-Sum Theorems

Examples: Identify the following as equilateral, equiangular or regular.

1) 2) 3)

Using Variables:

S = 180(n – 2) and I=180(n−2)n

Examples: Find the sum of the interior angles of each polygon. Then find the measure of each interior angle.4) Decagon 6) Heptagon 7) 15-gon

Examples: The sum of the angle measures of a polygon with n sides is given. Find n.8) 900 9) 1440

Example: Find the missing variables.

10)

What is special about the value of the interior angle and exterior angle at the same vertex?

Using Variables

E=360n

∧n=360E

∧I+E=180

Examples: Find the measure of an exterior angle of each regular polygon.

11) 12-gon 13) 24-gon

Examples: Find the number of sides of a regular polygon given the measure of the exterior angle.14) 20

Example: Find the number of sides of a regular polygon with an interior angle measure given.15) 144

6.2: Properties of Parallelograms

Parallelogram:

Opposite Sides:

Opposite Angles:

DiagramsDraw a diagram to model each of the theorems mentioned above.

Examples: Find the variable in the following figures.

1) 2)

3) 4)

What is true about BD and DF?

Examples: In the figure, GH = HI = IJ. Find each length.

5. EB 6. BD

7. AF 8. AK

9. CD 10. GJ

11. Complete a two-column proof.

Given: QRST, TSVU

Prove:

6.3: Proving that a Quadrilateral is a Parallelogram

Examples: Write P if the statement describes a parallelogram or appears to be a parallelogram. Write N if it does not. Explain your reasoning.

1) 5 congruent sides 2) Regular Quadrilateral 3) 4)

HOW DO WE PROVE THAT A QUADRILATERAL IS A PARALLELOGRAM?

DIAGRAMS: Model each theorem above on the given quadrilaterals.

Examples: Find the values of the variables that must make each quadrilateral a parallelogram.

5) 6)

7) 8)

Examples: Are the following parallelograms? If so, state the theorem that justifies it. If not, write not possible.9) 10) 11) 12)

13) Prove the following.

D

FG

E

H

6.4: Properties of Rhombuses, Rectangles and Squares

Examples: Complete each statement with always, sometimes or never.

DIAGRAMS:

Examples: Decide whether the parallelogram is a rhombus, a rectangle, or a square. Explain \

1. 2. 3. 4.

Examples: Find the measure of each numbered angle in the rhombus.

5. 6.

Examples: QRST is a rectangle. Find the value of x and the length of each diagonal.

7. QS x and RT = 6x 10 8. QS 5x + 12 and RT 6x 2

6.5: Conditions for Rhombuses, Rectangles and Squares

Draw a polygon that has no diagonals. Draw a polygon that has 2 diagonals.

Draw all of the diagonals from one vertex in the polygon.

Theorem 6-18 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Examples: Can you conclude that the parallelogram is a rhombus, rectangle, square or none. Explain.

1) 2) 3)

Examples: Find the value of x that makes the special parallelogram.4) rectangle 5) rhombus 6) square

7) rectangle 8) rhombus 9) rhombus

10) rectangle 11) rectangle 12) rhombus

6.6: Trapezoids and Kites

Midsegment of a Trapezoid:

Examples: Find the measure of the numbered angles or the value of the variable.1) 2) 3) AC = x +5; BD = 2x - 7

4) 5)

Kite: A quadrilateral with of consecutive sides that are and no opposite sides .

Examples: Find the measures of the numbered angles inside each kite.

6) 7) 8)

Examples: Find the values of the variables in each.

9) 10)

6.7: Polygons in a Coordinate Plane

You can classify figures in a coordinate plane by using formulas and characteristics we have learned.

Classifying Triangles

Example 1: Is the triangle with vertices A(0,1), B(4,4) and C(7,0) scalene, isosceles or equilateral.

Classifying Parallelograms:Example 2: Is a quadrilateral with vertices M(0,1), N(-1,4), (P(2,5) and Q(3,2) a rectangle, square or both?

x

y

Example 3: A quadrilateral has vertices What special quadrilateral is formed by connecting the midpoints of the sides?

6.8: Applying Coordinate Geometry

Sometimes variables are used as coordinates. Apply your techniques of the coordinate plane as well as formulas we have learned to find missing values.

Example: A rectangle is placed in a convenient position in the first quadrant of a coordinate plane. What is the missing label for the vertex?

Example: The vertices of the trapezoid are the origin along with A(4a, 4b), B(4c, 4b), and C(4d, 0). Find the midpoint of the midsegment of the trapezoid.

(0, 0)

A B

C x

y

Example: For the parallelogram, find coordinates for P without using any new variables.

x

y

(0,0) (b,0)

(_?_ , _?_)(0,a)

P

c0

(a, b)

x

y