Download Chapter 6 Notes - cloudfront.net

Chapter 6 Notes
Mrs. Myers – Geometry
Name ______________________________
Period ______
6.1 Polygons

Polygons = is a plane figure that
o Is formed by 3 or more segments called sides (no curves)
o Each side intersects exactly 2 other sides (no gaps and no crossing).

Vertex – each endpoint of a side of the polygon (vertices)
Ex. 1 Identifying polygons

Polygons are named by the number of sides they have:
# of sides
Type of polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon


Convex – when extending the sides of a polygon, the sides do not go through the
polygon (or the interior).
Nonconvex (Concave) – when extending the sides, it does go through the interior.
Ex. 2 Identifying convex or concave polygons.
A)
B)



Equilateral = all sides of a polygon are congruent.
Equiangular = all interior angles of a polygon are congruent.
Regular = if a polygon is equilateral and equiangular.
Ex. 2 Identify regular polygons
A)
B)
C)
* Diagonal = of a polygon is a segment that joins 2 nonconsecutive vertices.
* Theorem 6.1: Interior Angles of a Quadrilateral = the sum of the measures of the
interior angles of a quadrilateral is 360 .
m 1  m 2  m 3  m 4  360
Ex. 4 Find the missing angle measurements.
A) Find m Q and m R .
P
80
70
x
Q
S
2x
R
B) Find m F , m G, and m H .
C) Find m K , m L, m M . Is the quadrilateral JKLM regular?
6.2 Properties of Parallelograms

Parallelogram:

Theorem 6.2: Opposite sides are 

Theorem 6.3: Opposite angles are 

Theorem 6.4: Consecutive angles are supplementary.

Theorem 6.5: Diagonals bisect each other.
PQRS
Ex. 1 Find JH and LH if GHJK .
Ex. 2 Given ABCD . Find m A & m D .
Ex. 3 Given WXYZ . Find x.
A)
B)
6.3 Proving Quadrilaterals are Parallelograms
Ways to Prove a shape is a Parallelogram:
1. Show that BOTH pair of opposite sides are parallel ( Definition of
parallelogram).
2. Show that BOTH pair of opposite sides are congruent.
3. Show that BOTH pair of opposite angles are congruent.
4. Show that ONE angle is supplementary to both consecutive angles.
5. Show that the diagonals BISECT each other.
6. Show that ONE pair of opposite sides are congruent and parallel.
Ex. 1 Are you given enough information to determine whether the quadrilateral is a
parallelogram? Explain.
A)
B)
C)
D)
E)
F)
Ex. 2 What value for “x” and “y” will make the quadrilateral a parallelogram?
A)
B)
C)
6.4 Rhombuses, Rectangles, and Squares

3 Types of Parallelograms:
1. Rhombus = 4  sides.
2. Rectangle = 4 Right
3.
's .
Square = 4  sides AND 4 Right
Ex. 1 PQRS is a rhombus. What is y?
Ex. 2 MNOP is a rectangle. Find x.
's .
* Theorem 6.11: A rhombus has  diagonals. AC  BD
* Theorem 6.12: Diagonals of a rhombus bisect the angles.
* Theorem 6.13: The diagonals of a rectangle are  . AC  BD
6.5 Trapezoids and Kites
* Trapezoid = a quadrilateral with exactly one pair of parallel sides (bases).
A & B are base angles
Where
C & D are base angles
* Isosceles Trapezoid = legs are  of a trapezoid.
* Theorem 6.14: an isosceles trapezoid have  base angles.
A
B and
* Theorem 6.15:  base angles implies an isosceles trapezoid.
* Theorem 6.16: a trapezoid is isosceles if and only if its diagonals are  .
( AD  BC  AC  BD )
C
D.
Ex. 1 CDEF is an isosceles trapezoid with CE  10 and m E  95
Find
DF  ______
m C  _____
m D  _____
m F  _____
* Midsegment = is the segment that connects the midpoints of its legs.
* Theorem 6.17: Midsegment Theorem for Trapezoids = the midsegment of a trapezoid
1
is to each base and its length is the sum of the lengths of the bases.
2
1
MN AD BC , MN   AD  BC 
2
Ex. 2 Find AB.
* Kite: is a quadrilateral that has 2 pairs of consecutive  sides, but opposite sides are
NOT  .
* Theorem 6.18: a kite has  diagonals ( AC  BD )
* Theorem 6.19: a kite has exactly one pair of opposite angles that are 
( A  C ONLY !)
Ex. 3 GHJK is a kite. Find HP.
Ex. 4 RSTU is a kite.
Find
m R  _______
m S  _______
m T  _______
S
R
(x + 30)
125
U
x
T
6.7Areas of Triangles and Quadrilaterals
* Postulate 22: Area of a Square
A  s2
* Postulate 23: Area 
2  polygons have the same area.
* Postulate 24: Area Addition
The area of a region is the sum of the areas of its nonoverlapping parts.
* Theorem 6.20: Area of a Rectangle
A  BH
* Theorem 6.21: Area of a Parallelogram
A  BH
* Theorem 6.22: Area of a Triangle
A
BH
2
Ex. 1 Find the area of RSTU .
Ex. 2 Find the area
Ex. 3 Find the area of RST
Ex. 4 Find the area
Ex. 5 Rewrite the formula in terms of B.
A
1
BH
2
* Theorem 6.23: Area of a Trapezoid
A
H  B1  B2 
2
* Theorem 6.24: Area of a kite (where D1 & D2 represent the lengths of the diagonals).
A
D1 D2
2
* Theorem 6.25: Area of a rhombus (where D1 & D2 represent the lengths of the
diagonals).
A
D1 D2
2
Ex. 6 Find the area of trapezoid EFGH.
F
E
4
4
6
H
G
Ex. 7 Find the area of a kite
B
8
16
A
8
8
D
Ex. 8 Find the area.
10 in
3 in
8 in
16 in
C