Download Geo Quadrilaterals Notes 2014

Geometry
Name: ______________________________________
Geometry: Unit 5
Quadrilaterals
Topics Covered:
- Quadrilaterals
- Parallelograms
- Rectangles
- Squares
- Rhombuses
- Kites
- Trapezoids
- Polygon Sum Conjecture
Objectives:
- Identify and differentiate quadrilaterals
- Apply properties of special parallelograms
- Apply properties of parallelograms
- Apply properties of kites
- Apply properties of trapezoids
- Determine the sum of the angles of a polygon
- Determine the sum of the exterior angles of a polygon
Assignments:
Assignment Section
#
22
5.1
5.2
23
1.6
5.5
24
5.6
25
5.3
Quad
Rev
26
5.7
13.4
27
Page # and Problems
Date
Assigned
Date
Due
Pg. 259 #1, 3, 4, 5, 6, 7, 8, 13
Pg. 263 #1, 2, 4, 5, 6, 7, 8, 9, 10
Pg. 66 #1-10
Pg. 283 #1, 2, 3, 4, 5, 6
Pg. 294 #1-13
Pg. 271 #1-8
Pg. 304 #7, 8, 9, 15
Quadrilateral Proofs Worksheet
Unit 5 Quadrilaterals Study Guide
1
Geometry
Name: ________________________________
5.1/5.2 Polygon Angle Theorems
Investigation 1:
Polygon
Number of Sides
Number of Triangles
Sum of Angles
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
n-gon
The sum of the measures of the n interior angles of a n-gon is _____________________.
Regular polygon:
The measure of an interior angle of a REGULAR n-gon is ______________________.
Ex. What is the sum of the measures of a octagon? of a 20-gon?
Ex. The sum of the angles of a polygon is 1620o, how many sides does the polygon have?
Ex. What is the measure of an interior angle of a REGULAR decagon? of a REGULAR 13-gon?
Ex. Find the value of x.
Ex. Find the value of y.
2
Investigation 2:
Find the values of a, b and c.
Find the values of d, e, f and g.
For any polygon, the sum of the measures of a set of exterior angles is __________________.
Ex. Find the values of h, j and k.
Ex. Find the value of x.
The measure of an exterior angle of a REGULAR n-gon is _____________.
Ex. What is the measure of an exterior angle of a REGULAR heptagon? of a REGULAR octagon?
Ex. Find the number of sides of a regular polygon whose exterior angles each measure 10o? whose
exterior angles each measure 72o?
3
Geometry
Name: _________________________
5.6 Properties of Parallelograms
With a partner, match the term with the example.
Term
_____ Bisect
Example
_____ Complementary
(A)
Angles
(B)
(C)
(E)
(F)
_____ Congruent Angles
_____ Congruent Sides
60
30
(D)
_____ Diagonal
B
A
C
_____ Parallel
_____ Perpendicular
_____ Opposite Angles
(G)
(H)
Def: Quadrilateral
Draw 2 examples of quadrilaterals.
Draw 2 non-examples of quadrilaterals.
Def: Parallelogram
Properties:

__________________________
C
_____________________ are congruent
A
S


C
T
S
Two pairs of _____________ sides
C
A

A
T
_____________________ are
supplementary
C
S
A
T
S

_____________________ are congruent
C

T
Diagonals ______________ each other
C
A
A
D
S
T
S
T
4
Example: DOGS is a parallelogram.
D
List all of the congruent parts.
O
C
S
G
List the parallel sides.
List the supplementary angles.
Examples:
1. STUV is a parallelogram.
What is the relationship between ∠𝑆 𝑎𝑛𝑑 ∠𝑈?
___________________
Find the value of y.
S
T
42
3x + 2
6y
V
U
What is the relationship between ∠𝑆 𝑎𝑛𝑑 ∠𝑇? ____________________
Find the value of x.
2. The polygon is a parallelogram.
The diagonals of a parallelogram _________ each other.
Find the value of x.
9y + 1
2x
x+7
4
Find the value of y.
3. Find the value of n.
3n
15
n+8
21
5
Geometry
Name: __________________________________
5.6 Special Parallelograms
Use the following terms to complete the sentences. Terms may be used more than once or not at all.
Acute
Diagonal
Right
Adjacent
Four
Supplementary
Bisect
Opposite
Three
1. A __________________ is a polygon with 4
sides and 4 angles.
2. A __________________ is not a quadrilateral
because it has 3 sides.
3. A parallelogram has ____________________
pairs of parallel sides.
4. The opposite sides of a parallelogram are
__________________.
5. The opposite angles of a parallelogram are
__________________.
Congruent
Consecutive
Quadrilateral
Regular
Triangle
Two
6. A pair of __________________ angles are
supplementary in a parallelogram.
7. Diagonals of a parallelogram
__________________ each other.
8. Perpendicular lines form a
__________________ angle.
9. A __________________ polygon means all
sides are congruent and all angles are
congruent.
10. A __________________ connects two nonadjacent vertices of a polygon.
Special Parallelograms- Rhombus, Rectangle and Square
These quadrilaterals have all the properties of parallelograms and more.
Rhombus


____________________________
Four _______________ sides

Diagonals ______________ pair of opposite
angles
F
F
R
R
S
G
G
O

O
Diagonals are ______________________
F

Diagonal forms two _________________
triangles
R
F
S
R
S
G
O
G
O
6
Practice Problems:
1) The following is a rhombus.
Find the values of a, b, c, d
and e.
e
2) The following is a
rhombus. Find the values
of a, b, c and d.
3) TOAD is a rhombus.
𝑇𝑂 = 3𝑥 − 6 𝑎𝑛𝑑 𝐴𝐷 = 18. Find
the value of x. What is the
perimeter of TOAD?
d
c
b
c
d
a
a
110
32
O
b
T
A
D
_______________________________________________  _______________________________________________
Rectangles


________________________
Four _____________ angles

Diagonals are _______________
G
O
G
O
T
A
T
A
Practice Problems:
1) Given RECT. 𝑅𝐶 = 5𝑥 − 2 𝑎𝑛𝑑 𝐸𝑇 = 2𝑥 + 19.
Find the value of x. Find the length of RC and
ET.
R
E
T
C
2) BIRD is a rectangle. 𝐵𝐼 = 5𝑥 𝑎𝑛𝑑 𝑅𝐷 = 7𝑥 + 12.
Find the value of x. Find the length of BI and RD.
________________________________________  ____________________________________
7
Squares


____________________________
Four ____________ angles
P
I
S
G


Four _____________ sides
P
I
S
G
A ________________ quadrilateral
Practice Problem: SQUA is a square.
a) Find the value of y.
S
Q
b) Find the perimeter.
2y + 1
3y
15x
c) Find the value of x.
A
U
8
Geometry
Name: __________________________
5.3 Kites and Trapezoids
Sketch the following:
a) Isosceles triangle TRI where 𝑅𝐼 ≅ 𝐼𝑇.
c) A square with diagonals YU and OR. Mark
congruent sides and right angles.
b) Rhombus RHOM. Mark the parallel sides,
congruent sides and congruent angles.
d) A quadrilateral with one pair of parallel
sides.
S
Kites
E
L
A


_____________________
Two pairs of ____________ adjacent sides

Non-vertex angles are _______________
S
E
S
E
L
L
A
A

____________ are perpendicular

Diagonals ___________ vertex angles
S
E
S
E
L
L
A
A
9
Practice Problems
1) For the given kite, find the values of a, b, c, d and e.
b 60
a c
d
e
2) For the given kite:
a) Find the value of x.
7y + 2
105
6
b) Find the value of y.
z
16
c) Find the perimeter of the kite.
8 - 2x
d) Find the value of z.
____________________________________  ____________________________________
Trapezoids
F
I
S
H



_________________
______ pair of parallel sides
F
H
____________________ (non-base pair) are
supplementary
F
I
S
H
I
S
10
Practice Problems:
1) For the given trapezoid, find the value of x and y.
85 5x
y - 3 89
2) For trapezoid TRAP, ∠𝑇 = 80° 𝑎𝑛𝑑 ∠𝑅 = 40° (∠𝑇 𝑎𝑛𝑑 ∠𝑅 𝑎𝑟𝑒 𝑎 𝑝𝑎𝑖𝑟 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒𝑠). Find the
measures of ∠𝐴 𝑎𝑛𝑑 ∠𝑃?
___________________________________  ______________________________________
Isosceles Trapezoids

A trapezoid with _________ legs

________________ are congruent
E
B
B
A
R
R
Practice Problems:
1) Find the values of a and b.
141
b
E
A
2) For trapezoid ABCD, where 𝐴𝐵 ∥ 𝐶𝐷, 𝐴𝐶 =
35, 𝐷𝐵 = 5𝑦, 𝐴𝐷 = 10 𝑎𝑛𝑑 𝐵𝐶 = 2 − 𝑥, find
the values of x and y.
a
11
Quadrilateral Flow-Chart
12
Geometry
Name: _______________________________
5.7 Proving Quadrilateral Properties
Circle if the following statements are TRUE or FALSE
TRUE/FALSE The diagonals of a kite are perpendicular.
TRUE/FALSE A rectangle is a rhombus.
TRUE/FALSE A trapezoid has two pairs of parallel sides.
TRUE/FALSE The diagonals of a rectangle bisect each other.
TRUE/FALSE A rhombus is a regular quadrilateral.
TRUE/FALSE A consecutive angles of a parallelogram are supplementary.
What is a geometric proof?
Using already proven conjectures, axioms, postulates and theorems to prove hypothesis and
statements.
EXAMPLE 1
Prove: The diagonal of a parallelogram divides the parallelogram
into two congruent triangle.
̅̅̅̅.
Given: Parallelogram SOAK with diagonal 𝑆𝐴
Show: ∆𝑆𝑂𝐴 ≅ ∆𝐴𝐾𝑆
Statement
Reason
SOAK is a parallelogram
̅̅̅̅
𝑆𝑂 ∥ ̅̅̅̅
𝐾𝐴
̅̅̅̅
𝑂𝐴 ∥ ̅̅̅̅
𝑆𝐾
∠3 ≅ ∠4
∠1 ≅ ∠2
̅̅̅̅
𝑆𝐴 ≅ ̅̅̅̅
𝑆𝐴
∆𝑆𝑂𝐴 ≅ ∆𝐴𝐾𝑆
13
Example 2
Prove: The diagonals of a rectangle are congruent
Given: Rectangle YOGI with diagonals ̅̅̅̅
𝑌𝐺 𝑎𝑛𝑑 ̅̅̅
𝑂𝐼
̅̅̅̅
̅̅̅
Show: 𝑌𝐺 ≅ 𝑂𝐼
Statement
Reason
Rectangle YOGI with
diagonals ̅̅̅̅
𝑌𝐺 and ̅̅̅
𝑂𝐼
̅̅̅
𝐼𝑌 ≅ ̅̅̅̅
𝐺𝑂
∠𝑌𝑂𝐺 ≅ ∠𝑂𝑌𝐼
̅̅̅̅
𝑌𝑂 ≅ ̅̅̅̅
𝑂𝑌
∆𝑌𝑂𝐺 ≅ ∆𝑂𝑌𝐼
̅̅̅̅
𝑌𝐺 ≅ ̅̅̅
𝐼𝑂
Use the following terms to complete the proof:
ASA
Definition of rectangle
Definition of parallelogram
Given
Opposite sides of rectangle are ≅
Same segment CPCTC
SAS
CPCTC (Corresponding Parts of Congruent Triangles are
Congruent)
Example 3
Prove: The diagonals of an isosceles trapezoid are congruent
Given: Isosceles trapezoid GTHR with ̅̅̅̅
𝐺𝑅 ≅ ̅̅̅̅
𝑇𝐻 and diagonals ̅̅̅̅
𝐺𝐻 and
̅̅̅̅
𝑇𝑅
̅̅̅̅ ≅ 𝑇𝑅
̅̅̅̅
Show: 𝐺𝐻
Statement
Reason
Isosceles trapezoid GTHR
̅̅̅̅ ≅ 𝑇𝐻
̅̅̅̅
𝐺𝑅
∠𝑅𝐺𝑇 ≅ ∠𝐻𝑇𝐺
̅̅̅̅ ≅ 𝐺𝑇
̅̅̅̅
𝐺𝑇
∆𝑅𝐺𝑇 ≅ ∆𝐻𝑇𝐺
̅̅̅̅ ≅ 𝑇𝑅
̅̅̅̅
𝐺𝐻
Use the following terms to complete the proof:
ASA
CPCTC
Base angles of isosceles trapezoid are ≅
Given
Given
Parallel lines
Same segment
SAS
14
Diagonals bisect each other
Diagonals are congruent
Diagonals bisect the angles
Diagonals are perpendicular
Diagonals bisects vertex angles
4 right angles
All sides are congruent
Consecutive angles are
supplementary
Non-vertex angles are
congruent
One pair of opposite sides are
congruent
Opposite sides are parallel
Opposite angles are
congruent
Consecutive angles are
congruent
Base angles are congruent
Two pairs of parallel sides
One pair of parallel sides
Quadrilaterals
Put an X in the box, if the quadrilateral has that property.
Parallelogram
Rectangle
Rhombus
Square
Kite
Trapezoid
Isosceles Trapezoid
15
Quadrilaterals
Parallelogram
Rectangle
Rhombus
2 in
3.1 in
1.3 in
2 in
1.3 in
2 in
3.1 in
3.1 in
1.3 in
2 in
1.3 in
3.1 in
Square
Kite
Trapezoid
Isosceles
Trapezoid
16