Ch. 6 Notes
6.1: 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=
𝟑𝟔𝟎
𝟑𝟔𝟎
𝒂𝒏𝒅 𝒏 =
𝒂𝒏𝒅 𝑰 + 𝑬 = 𝟏𝟖𝟎
𝒏
𝑬
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:
Diagrams
Draw 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,
Prove: RQ  VU
TSVU
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.
Given: HGD  HEF
Prove: DEFG is a parallelogram
D
E
H
G
F
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
and no opposite sides
of consecutive sides that are
.
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?
Example 3: A quadrilateral has vertices
quadrilateral is formed by connecting the midpoints of the sides?
What special
y
x
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?
y
(0,a)
(0,0)
(_?_ , _?_)
(b,0)
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.
y
A
B
(0, 0)
C
x
Example: For the parallelogram, find coordinates for P without using any new variables.
y
(a, b)
0
P
c
x
x