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