Transcript 6.1 Polygons - John C. Fremont High School

6.1 Polygons
Day 1 Part 1
CA Standards 7.0, 12.0, 13.0
Warmup

Solve for the variables.

1. 10 + 8 + 16 + A = 36

2. 6 + 15 + 9 + 3B = 36

3. 10 + 8 + 2X + 2X = 36

4. 4R + 10 + 108 + 67 + 3R = 360
What is polygon?

Formed by three or more segments (sides).

Each side intersects exactly two other sides,
one at each endpoint.

Has vertex/vertices.

Polygons are named by the number of sides they
have. Fill in the blank.
Number of sides
Type of polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
Concave vs. Convex

Convex: if no line that contains a side of
the polygon contains a point in the interior
of the polygon.
interior

Concave: if a polygon is not convex.
Example

Identify the polygon and state whether it is
convex or concave.
Concave polygon
Convex polygon



A polygon is equilateral if all of its sides
are congruent.
A polygon is equiangular if all of its interior
angles are congruent.
A polygon is regular if it is equilateral and
equiangular.
Decide whether the polygon is regular.
))
)
))
)

A Diagonal of a polygon is a segment that
joins two nonconsecutive vertices.
diagonals
Interior Angles of a Quadrilateral
Theorem

The sum of the measures of the interior
angles of a quadrilateral is 360°.
B
m<A + m<B + m<C + m<D = 360°
C
A
D
Example

Find m<Q and m<R.
x + 2x + 70° + 80° = 360°
3x + 150 ° = 360 °
3x = 210 °
x = 70 °
Q
x
R
m< Q = x
m< Q = 70 °
2x°
80° P
70°
S
m<R = 2x
m<R = 2(70°)
m<R = 140 °
Find m<A
C
65°
D
55°
123°
B
A

Use the information in the diagram to solve
for j.
60° + 150° + 3j ° + 90° = 360°
60°
210° + 3j ° + 90° = 360°
150°
300° + 3j ° = 360 °
3j °
3j ° = 60 °
j = 20
6.2 Properties of Parallelograms
Day 1 Part 2
CA Standards 4.0, 7.0, 12.0, 13.0,
16.0, 17.0
Theorems

If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Q
PQ  RS
SP  QR
P

R
S
If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
P  R
Q  S
Theorems

If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
m<P + m<Q = 180°
m<Q + m<R = 180°
m<R + m<S = 180°
m<S + m<P = 180°
Q
P
R
S
Using Properties of Parallelograms

PQRS is a parallelogram. Find the angle
measure.


Q
m< R 70 °
m< Q
70 ° + m < Q = 180 °
70°
P
m< Q = 110 °
R
S
Using Algebra with Parallelograms

PQRS is a parallelogram. Find the value
of h.
P
3h
S
Q
120°
R
Theorems

If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
R
Q
QM  SM
PM  RM
M
P
S
Using properties of parallelograms

FGHJ is a parallelogram. Find the
unknown length.


JH 5
JK 3
5
F
G
3
K
J
H
Examples

Use the diagram of parallelogram JKLM.
Complete the statement.
LM
1.JK  ____
NK
2.MN  ____
K
3.MLK  <KJM
____
4.JKL  ____
<LMJ
5.JN  ____
NL
6.KL  ____
MJ
L
N
J
M
Find the measure in parallelogram
LMNQ.
LM 18
2. LP 8
3. LQ 9
4. QP 10
5. m<LMN 70°
6. m<NQL 70 °
7. m<MNQ 110 °
8. m<LMQ 32 °
1.
L
110°
M
10
9
P
8
32°
Q
18
N


Pg. 325 # 4 – 20, 24 – 34, 37 – 46
Pg. 333 # 2 – 39
6.3 Proving Quadrilaterals are
Parallelograms
Day 2 Part 1
CA Standards 4.0, 7.0, 12.0, 17.0
Warmup

Find the slope of AB.

A(2,1), B(6,9)
m=2

A(-4,2), B(2, -1)
m= - ½

A(-8, -4), B(-1, -3)
m= 1/7
Review
rise
y2  y1
slope 

run
x2  x1
d
x2  x1 
2
  y2  y1 
2
Using properties of parallelograms.



Method 1
Use the slope formula to show that
opposite sides have the same slope, so
they are parallel.
Method 2
Use the distance formula to show that the
opposite sides have the same length.
Method 3
Use both slope and distance formula to
show one pair of opposite side is
congruent and parallel.
Let’s apply~

Show that A(2,0), B(3,4), C(-2,6), and D(3,2) are the vertices of parallelogram by
using method 1.

Show that the quadrilateral with vertices
A(-3,0), B(-2,-4), C(-7, -6) and D(-8, -2) is a
parallelogram using method 2.

Show that the quadrilateral with vertices
A(-1, -2), B(5,3), C(6,6), and D(0,7) is a
parallelogram using method 3.
Proving quadrilaterals are
parallelograms




Show that both pairs of opposite sides are
parallel.
Show that both pairs of opposite sides are
congruent.
Show that both pairs of opposite angles
are congruent.
Show that one angle is supplementary to
both consecutive angles.
.. continued..


Show that the diagonals bisect each other
Show that one pair of opposite sides are
congruent and parallel.

Show that the quadrilateral with vertices
A(-1, -2), B(5,3), C(6,6), and D(0,7) is a
parallelogram using method 3.

Show that A(2,-1), B(1,3), C(6,5), and
D(7,1) are the vertices of a parallelogram.
6.4 Rhombuses, Rectangles, and
Squares
Day 2 Part 2
CA Standards 4.0, 7.0, 12.0, 17.0
Review

Find the value of the variables.
p
h
52°
(2p-14)°
68°
52° + 68° + h = 180°
120° + h = 180 °
h = 60°
50°
p + 50° + (2p – 14)° = 180°
p + 2p + 50° - 14° = 180°
3p + 36° = 180°
3p = 144 °
p = 48 °
Special Parallelograms

Rhombus

A rhombus is a parallelogram with four
congruent sides.
Special Parallelograms

Rectangle

A rectangle is a parallelogram with four right
angles.
Special Parallelogram

Square

A square is a parallelogram with four
congruent sides and four right angles.
Corollaries

Rhombus corollary


Rectangle corollary


A quadrilateral is a rhombus if and only if it has
four congruent sides.
A quadrilateral is a rectangle if and only if it
has four right angles.
Square corollary

A quadrilateral is a square if and only if it is a
rhombus and a rectangle.
Example

PQRS is a rhombus. What is the value of
b?
2b + 3 = 5b – 6
9 = 3b
3=b
Q
P
2b + 3
S
5b – 6
R
Review

In rectangle ABCD, if AB = 7f – 3 and CD
= 4f + 9, then f = ___
1
B) 2
C) 3
D) 4
E) 5
A)
7f – 3 = 4f + 9
3f – 3 = 9
3f = 12
f=4
Example

PQRS is a rhombus. What is the value of
b?
3b + 12 = 5b – 6
18 = 2b
9=b
Q
P
3b + 12
S
5b – 6
R
Theorems for rhombus

A parallelogram is a rhombus if and only if
its diagonals are perpendicular.

A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite
angles.
Theorem of rectangle

A parallelogram is a rectangle if and only if
its diagonals are congruent.
A
D
B
C
Match the properties of a quadrilateral
1.
2.
3.
4.
5.
6.
The diagonals are
congruent B,D
Both pairs of opposite
sides are congruent A,B,C,D
Both pairs of opposite
sides are parallel A,B,C,D
All angles are congruent B,D
All sides are congruent C,D
Diagonals bisect the
angles C
Parallelogram
B. Rectangle
C. Rhombus
D. Square
A.
6.5 Trapezoid and Kites
Day 3 Part 1
CA Standards 4.0, 7.0, 12.0
Warmup

Which of these sums is
equal to a negative
number?
A)
(4) + (-7) + (6)
(-7) + (-4)
(-4) + (7)
(4) + (7)
B)
C)
D)

A)
B)
C)
D)
In the first seven games
of the basketball
season, Cindy scored 8,
2, 12, 6, 8, 4 and 9
points. What was her
mean number of points
scored per game?
6
7
8
9
Let’s define Trapezoid
A
base
>
leg
D
B
leg
>
base
C
<D AND <C ARE ONE PAIR OF BASE ANGLES.
When the legs of a trapezoid are congruent,
then the trapezoid is an isosceles trapezoid.
Isosceles Trapezoid

If a trapezoid is isosceles, then each pair of
base angles is congruent.
A
D
A  B, C  D
B
C
PQRS is an isosceles trapezoid. Find
m<P, m<Q, and m<R.
S
50°
P
R
>
>
Q
Isosceles Trapezoid

A trapezoid is isosceles if and only if its
diagonals are congruent.
A
B
AC  BD
D
C
Midsegment Theorem for
Trapezoid

The midsegment of a trapezoid is parallel to
each base and its length is one half the sum of
the lengths of the base.
B
M
1
MN  ( AD  BC )
2
A
C
N
D
Examples
 The
midsegment of the trapezoid is RT.
Find the value of x.
7
R
x
14
T
x = ½ (7 + 14)
x = ½ (21)
x = 21/2
Examples
 The
midsegment of the trapezoid is ST.
Find the value of x.
8
S
11
x
T
11 = ½ (8 + x)
22 = 8 + x
14 = x
Review
In a rectangle ABCD, if AB = 7x – 3, and CD =
4x + 9, then x = ___
A) 1
B) 2
C) 3
D) 4
E) 5
7x – 3 = 4x + 9
-4x
-4x
3x – 3 =
9
+3
+3
3x = 12
x = 4
Kite
 A kite
is a quadrilateral that has two pairs of
consecutive congruent sides, but opposite
sides are congruent.
Theorems about Kites

If a quadrilateral is a kite, then its diagonals are
perpendicular

If a quadrilateral is a kite, then exactly one pair of
opposite angles are congruent.
A  C, B  D
A
B
L
D
C
Example
 Find
m<G and m<J.
J
Since m<G = m<J,
2(m<G) + 132° + 60° = 360°
2(m<G) + 192° = 360°
2(m<G) = 168°
60° K
m<G = 84°
H 132°
G
Example
 Find
the side length.
J
12
H
12
14
12
G
K
6.6 Special Quadrilaterals
Day 3 Part 2
CA Standards 7.0, 12.0
Summarizing Properties of
Quadrilaterals
Quadrilateral
Kite
Parallelogram Trapezoid
Isosceles Trapezoid
Rhombus
Rectangle
Square
Identifying Quadrilaterals

Quadrilateral ABCD has at least one pair of
opposite sides congruent. What kinds of
quadrilaterals meet this condition?


Sketch KLMN. K(2,5), L(-2,3), M(2,1),
N(6,3).
Show that KLMN is a rhombus.
Copy the chart. Put an X in the box if the shape
always has the given property.
Property
Both pairs of
opp. sides are ll
Parallelo Rectangle Rhombus Square Kite Trapezoid
gram
X
X
X
X
Exactly 1 pair of
opp. Sides are ll
X
Diagonals are
perp.
X
Diagonals are
cong.
Diagonals
bisect each
other
X
X
X
X
X
X

Determine whether the statement is true or
false. If it is true, explain why. If it is false,
sketch a counterexample.

If CDEF is a kite, then CDEF is a convex
polygon.

If GHIJ is a kite, then GHIJ is not a trapezoid.

The number of acute angles in a trapezoid is
always either 1 or 2.


Pg. 359 # 3 – 33, 40
Pg. 368 # 16 – 41
6.7 Areas of Triangles and
Quadrilaterals
Day 4 Part 1
CA Standard 7.0, 8.0, 10.0
Warmup
1. 5  2 1 
6
5
2. 3
2


4 11
3. 11  1
1
   
12  3 4 
Area Postulates

Area of a Square Postulate


Area Congruence Postulate


The area of a square is the square of the
length of its sides, or A = s2.
If two polygons are congruent, then they have
the same area.
Area Addition Postulate

The area of a region is the sum of the areas of
its non-overlapping parts.
Area
 Rectangle: A =
bh
 Parallelogram: A = bh
 Triangle: A = ½ bh
 Trapezoid: A = ½ h(b1+b2)
 Kite: A = ½ d1 d2
 Rhombus: A = ½ d1 d2
the area of ∆ ABC.
C
7
4 6
L
 Find
A
5
B
 Find
the area of a trapezoid with vertices at
A(0,0), B(2,4), C(6,4), and D(9,0).
Find the area of the figures.
L
4
L
L
L
4
4
2
L
L
L
4
5
8
12
Find the area of ABCD.
B
C
9
E
16
A
12
D
ABCD is a parallelogram
Area = bh
= (16)(9)
= 144
Find the area of a trapezoid.
 Find
the area of a trapezoid WXYZ with
W(8,1), X(1,1), Y(2,5), and Z(5,5).
Find the area of rhombus.
 Find
the area of rhombus ABCD.
B
15
A
20
20
15
25
D
C
Area of
Rhombus
A = ½ d1 d2
= ½ (40)(30)
= ½ (1200)
= 600
 The
area of the kite is160.
 Find the length of BD.
A
D
10
C
B
Ch 6 Review
Day 4 Part 2
Review 1

A polygon with 7 sides is called a ____.
A)
B)
C)
D)
E)
nonagon
dodecagon
heptagon
hexagon
decagon
Review 2
 Find
A)
B)
C)
D)
E)
m<A
65°
135°
100°
90°
105°
B
A
165°
65°
D
30°
C
Review 3
 Opposite
angles of a parallelogram must be
_______.
A) complementary
B) supplementary
C) congruent
D) A and C
E) B and C
Review 4
 If
a quadrilateral has four equal sides, then
it must be a _______.
A) rectangle
B) square
C) rhombus
D) A and B
E) B and C
Review 5
 The
perimeter of a square MNOP is 72
inches, and NO = 2x + 6. What is the value
of x?
A) 15
B) 12
C) 6
D) 9
E) 18
Review 6
 ABCD
is a trapezoid. Find the length of
midsegment EF.
13
A) 5
A
B) 11
E
C) 16
11
B
D) 8
5
E) 22
D
F
9
C
Review 7
 The
quadrilateral below is most specifically
a __________.
A) rhombus
B) rectangle
C) kite
D) parallelogram
E) trapezoid
Review 8
 Find
the base length of a triangle with an
area of 52 cm2 and a height of 13cm.
A) 8 cm
B) 16 cm
C) 4 cm
D) 2 cm
E) 26 cm
Review 9
 A right
triangle has legs of 24 units and 18
units. The length of the hypotenuse is
____.
A) 15 units
B) 30 units
C) 45 units
D) 15.9 units
E) 32 units
Review 10
 Sketch
a concave pentagon.
 Sketch
a convex pentagon.
Review 11
 What
type of quadrilateral is ABCD?
Explain your reasoning.
D
120°
A
60°
C
120°
Isosceles Trapezoid
Isosceles : AD = BC
Trapezoid : AB ll CD
60°
B

Pg. 382 # 1 - 25