Geometry: Unit 5 Quadrilaterals - Cerrone's Websitebcerrone.com/EHS Website/Geometry/Geo Quadrilaterals Notes 2014... · -Apply properties of special ... Ex. Find the number of sides

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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.

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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

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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

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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°

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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

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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

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Quadrilaterals Put an X in the box, if the quadrilateral has that property.

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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