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1
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 Page # and Problems Date
Assigned Date Due
22 5.1 5.2
Pg. 259 #1, 3, 4, 5, 6, 7, 8, 13 Pg. 263 #1, 2, 4, 5, 6, 7, 8, 9, 10
23 1.6 5.5
Pg. 66 #1-‐10 Pg. 283 #1, 2, 3, 4, 5, 6
24 5.6 Pg. 294 #1-‐13 25 5.3
Quad Rev
Pg. 271 #1-‐8 Pg. 304 #7, 8, 9, 15
26 5.7 13.4
Quadrilateral Proofs Worksheet
27 Unit 5 Quadrilaterals Study Guide
2
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.
3
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?
4
S T
AC
S T
AC
S T
AC
S T
AC
S T
AC
D
S T
AC
Geometry Name: _________________________
5.6 Properties of Parallelograms
With a partner, match the term with the example.
Term Example _____ Bisect
_____ Complementary
Angles
_____ Congruent Angles
_____ Congruent Sides
_____ Diagonal
_____ Parallel
_____ Perpendicular
_____ Opposite Angles
(A) (B) (C)
(D) (E) (F)
(G) (H)
Def: Quadrilateral Draw 2 examples of quadrilaterals.
Draw 2 non-examples of quadrilaterals.
Def: Parallelogram Properties:
• __________________________
• _____________________ are congruent
• Two pairs of _____________ sides
• _____________________ are supplementary
• _____________________ are congruent
• Diagonals ______________ each other
60°
30° CBA
5
C
SG
OD
V U
TS3x + 2
6y
42°
Example: DOGS is a parallelogram.
List all of the congruent parts.
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.
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. Find the value of y.
3. Find the value of n.
4
9y + 1x + 72x
15n + 8
21
3n
6
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 Adjacent Bisect Congruent Consecutive Diagonal Four Opposite Quadrilateral Regular Right Supplementary Three Triangle Two
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
__________________.
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 non-adjacent vertices of a polygon.
Special Parallelograms- Rhombus, Rectangle and Square
These quadrilaterals have all the properties of parallelograms and more.
Rhombus
• ____________________________ • Four _______________ sides
• Diagonals are ______________________
• Diagonals ______________ pair of opposite angles
• Diagonal forms two _________________
triangles
GO
RF
S
GO
RF
S
GO
RF
S
GO
RF
7
Practice Problems:
1) The following is a rhombus. Find the values of a, b, c, d and 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?
_______________________________________________ J _______________________________________________
Rectangles
• ________________________ • Four _____________ angles
• Diagonals are _______________
Practice Problems:
1) Given RECT. 𝑅𝐶 = 5𝑥 − 2 𝑎𝑛𝑑 𝐸𝑇 = 2𝑥 + 19. Find the value of x. Find the length of RC and ET.
2) BIRD is a rectangle. 𝐵𝐼 = 5𝑥 𝑎𝑛𝑑 𝑅𝐷 = 7𝑥 + 12.
Find the value of x. Find the length of BI and RD.
________________________________________ J ____________________________________
110°da
bce
32°
d
c
ba
D
A
O
T
T A
OG
T A
OG
T C
ER
8
15x
2y + 1 3y
A U
QS
Squares
• ____________________________ • Four ____________ angles
• Four _____________ sides
• A ________________ quadrilateral
Practice Problem: SQUA is a square.
a) Find the value of y.
b) Find the perimeter.
c) Find the value of x.
S G
IP
S G
IP
9
L
A
ES
Geometry Name: __________________________
5.3 Kites and Trapezoids
Sketch the following:
a) Isosceles triangle TRI where 𝑅𝐼 ≅ 𝐼𝑇.
b) Rhombus RHOM. Mark the parallel sides, congruent sides and congruent angles.
c) A square with diagonals YU and OR. Mark congruent sides and right angles.
d) A quadrilateral with one pair of parallel sides.
Kites
• _____________________ • Two pairs of ____________ adjacent sides
• ____________ are perpendicular
• Non-vertex angles are _______________
• Diagonals ___________ vertex angles
L
A
ES
L
A
ES
L
A
ES
L
A
ES
10
Practice Problems 1) For the given kite, find the values of a, b, c, d and e.
2) For the given kite:
a) Find the value of x.
b) Find the value of y.
c) Find the perimeter of the kite.
d) Find the value of z.
____________________________________ C ____________________________________
Trapezoids
• _________________ • ______ pair of parallel sides
• ____________________ (non-base pair) are supplementary
60°
ed cba
H S
IF
H S
IF
z
105°
8 - 2x
6
16
7y + 2
H S
IF
11
Practice Problems:
1) For the given trapezoid, find the value of x and y.
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?
___________________________________ C ______________________________________
Isosceles Trapezoids
• A trapezoid with _________ legs
• ________________ are congruent
Practice Problems: 1) Find the values of a and b.
2) For trapezoid ABCD, where 𝐴𝐵 ∥ 𝐶𝐷,𝐴𝐶 =35,𝐷𝐵 = 5𝑦,𝐴𝐷 = 10 𝑎𝑛𝑑 𝐵𝐶 = 2 − 𝑥, find the values of x and y.
y - 3 89°
5x85°
R A
EB
R A
EB
b a
141°
12
Quadrilateral Flow-Chart
13
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
14
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 Show: 𝐺𝐻 ≅ 𝑇𝑅
Statement Reason
Isosceles trapezoid GTHR
𝐺𝑅 ≅ 𝑇𝐻
∠𝑅𝐺𝑇 ≅ ∠𝐻𝑇𝐺
𝐺𝑇 ≅ 𝐺𝑇
∆𝑅𝐺𝑇 ≅ ∆𝐻𝑇𝐺
𝐺𝐻 ≅ 𝑇𝑅
Use the following terms to complete the proof: ASA CPCTC Base angles of isosceles trapezoid are ≅ Given
Given Parallel lines Same segment SAS
15
Quadrilaterals Put an X in the box, if the quadrilateral has that property.
One
pai
r of
par
alle
l sid
es
Two
pai
rs o
f par
alle
l sid
es
Op
pos
ite
ang
les
are
cong
ruen
t C
onse
cuti
ve a
ngle
s ar
e co
ngru
ent
Bas
e an
gle
s ar
e co
ngru
ent
Con
secu
tive
ang
les
are
sup
ple
men
tary
N
on-v
erte
x an
gle
s ar
e co
ngru
ent
One
pai
r of
op
pos
ite
sid
es a
re
cong
ruen
t
Op
pos
ite
sid
es a
re p
aral
lel
All
sid
es a
re c
ong
ruen
t
4 ri
ght
ang
les
Dia
gon
als
bis
ects
ver
tex
ang
les
Dia
gon
als
are
per
pen
dic
ular
Dia
gon
als
bis
ect t
he a
ngle
s
Dia
gon
als
are
cong
ruen
t
Dia
gon
als
bis
ect e
ach
othe
r
Parallelogram
Rectangle
Rhombus
Square
Kite
Trapezoid
Isosceles Trapezoid
16
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square Kite
Trapezoid Isosceles Trapezoid
2 in
2 in
2 in
2 in
1.3 in
1.3 in
1.3 in
1.3 in
3.1 in
3.1 in3.1 in
3.1 in