1

Properties

of

Quadrilaterals

2

D C

BA

132x - 11

J

M

K

L -6 + 2x

x + 1T W

U V

Proving Properties of Parallelograms

Given: ABCD is a parallelogram Prove: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

Property 1: Opposite sides of a parallelogram are _________________________________.

Guided Practice: Solve for the missing variable. 1. Solve for x. 2. Find VU

Statements Reasons

1. 1. Given

2. 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ∥ 𝐵𝐶̅̅ ̅̅ 2.

3. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵 and ∠𝐶𝐵𝐷 ≅ ∠𝐴𝐷𝐵 3.

4. 𝐵𝐷̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ 4.

5. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵 5.

6. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ≅ 𝐶𝐵̅̅ ̅̅ 6.

3

A B

CD

1 + 19x

58°

M

N K

L

Given: ABCD is a parallelogram Prove: ∠𝐵 ≅ ∠𝐷 and ∠𝐴 ≅ ∠𝐶 (Hint: to make things easier, you may want to draw your triangles out to the side)

Property 2: Opposite angles of a parallelogram are _________________________________.

Guided Practice Solve for the missing variable. 1. Solve for x. 2. Solve for x.

Statements Reasons

1. 1. Given

2. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ 2.

3. 𝐵𝐷̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ and 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ 3.

4. ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴 and ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵 4.

5. 5. CPCTC

4

P

D C

BA

5x - 9

3x + 11E

DG

F

10x - 6

8x + 6

T S

U V

Find the measure indicated in each parallelogram. 3. Find m G 4. Find m V Given: ABCD is a parallelogram Prove: 𝐴𝑃̅̅ ̅̅ ≅ 𝑃𝐶̅̅̅̅ and 𝐷𝑃̅̅ ̅̅ ≅ 𝑃𝐵̅̅ ̅̅

Property 3: Diagonals of a parallelogram _____________________________ each other.

Statements Reasons

1. ABCD is a parallelogram 1.

2. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ 2.

3. 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ∥ 𝐵𝐶̅̅ ̅̅ 3.

4. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵 and ∠𝐶𝐴𝐵 ≅ ∠𝐴𝐶𝐷 4.

5. ∆𝐴𝑃𝐵 ≅ ∆𝐶𝑃𝐷 5.

6. 𝐴𝑃̅̅ ̅̅ ≅ 𝐶𝑃̅̅̅̅ and 𝐷𝑃̅̅ ̅̅ ≅ 𝐵𝑃̅̅ ̅̅ 6.

5

A B

CD

R

W

X Y

Z

N

F G

E D

A

ST

RU

Guided Practice Solve for the missing variable or measure indicated. 1. Solve for x. YR = 19 and RW = 3x – 2 2. Solve for x. NG = 19 and EG = 4x + 6

3. SA = x + 11 and AU = 2x + 1, Find SU

Given: ABCD is a parallelogram Prove: 𝑚∠𝐴 + 𝑚∠𝐷 = 180 and 𝑚∠𝐶 + 𝑚∠𝐷 = 180

Statements Reasons

1. ABCD is a parallelogram 1.

2. 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ∥ 𝐵𝐶̅̅ ̅̅ 2.

3. 𝑚∠𝐴 + 𝑚∠𝐷 = 180 3.

4. 𝑚∠𝐶 + 𝑚∠𝐷 = 180 4.

6

65°

20x - 5T

SV

U

8 + 26x

8x + 2

Z W

Y X

15x +5

19x + 5N

M

K

L

Property 4: Consecutive angles of a parallelogram are ______________________________.

Guided Practice: Solve for the missing variable or measurement indicated. 1. Find x. 2. 8 2m W x and 8 26m X x

Find m X

3. Find m N

7

Proving Properties of Rectangles

Property: Diagonals of a rectangle are __________________________________.

Guided Practice: Solve for x. 1. AC= 15 and BD= 2x + 7 2. AR= 3x, RC= 4, and BD= 25

T

P

S R

Q

Statements Reasons

1. PQRS is a rectangle. Given

2. PQRS is a parallelogram. A rectangle is a parallelogram.

3. QR SP

4. RS SR

5. m PSR = 90

mQRS = 90 Definition of a rectangle

6. m PSR = mQRS Transitive property

7. PRS PRS

8. PR QS

8

H

G

FE

D

Proving Properties of Rhombi Given: DEFG is a rhombus Prove: ∠𝐸𝐹𝐻 ≅ ∠𝐺𝐹𝐻 and ∠𝐷𝐸𝐻 ≅ ∠𝐹𝐸𝐻

Property: Diagonals of a rhombus ______________________________ opposite angles.

Statements Reasons

1. DEFG is a rhombus 1.

2. 𝐸𝐹̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅ ≅ 𝐺𝐷̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅ 2.

3. DEFG is a parallelogram 3.

4. 𝐸𝐻̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅ and 𝐷𝐻̅̅ ̅̅ ≅ 𝐻𝐹̅̅ ̅̅ 4.

5. ∆𝐸𝐻𝐹 ≅ ∆𝐺𝐻𝐹 5.

6. ∠𝐸𝐹𝐻 ≅ ∠𝐺𝐹𝐻 6.

7. ∆𝐷𝐸𝐻 ≅ ∆𝐹𝐸𝐻 7.

8. ∠𝐷𝐸𝐻 ≅ ∠𝐹𝐸𝐻 8.

9

H

G

FE

D

Given: DEFG is a rhombus Prove: 𝐸𝐺̅̅ ̅̅ ⊥ 𝐹𝐷̅̅ ̅̅

Property: Diagonals of a rhombus are __________________________________.

Since a square is a regular quadrilateral, what properties apply to a square?

Statements Reasons

1. DEFG is a rhombus 1.

2. 𝐸𝐹̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅ ≅ 𝐺𝐷̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅ 2.

3. DEFG is a parallelogram 3.

4. 𝐸𝐻̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅ and 𝐷𝐻̅̅ ̅̅ ≅ 𝐻𝐹̅̅ ̅̅ 4.

5. ∆𝐸𝐻𝐹 ≅ ∆𝐺𝐻𝐹 5.

6. ∠𝐸𝐻𝐹 ≅ ∠𝐺𝐻𝐹 6.

7. 𝑚∠𝐸𝐻𝐹 = 𝑚∠𝐺𝐻𝐹 7. Definition of Congruence

8. 𝑚∠𝐸𝐻𝐹 + 𝑚∠𝐺𝐻𝐹 = 180 8. Definition of Linear Pair

9. 𝑚∠𝐸𝐻𝐹 + 𝑚∠𝐸𝐻𝐹 = 180 9. Substitution

10. 2(𝑚∠𝐸𝐻𝐹) = 180 10.

11. 𝑚∠𝐸𝐻𝐹 = 90 11.

12. 𝑚∠𝐺𝐻𝐹 = 90 12.

13. 13. Definition of perpendicular

10

74°66°

3x + 4

DE

CF

13x - 2

80°50°T

U

S

R

5x + 5

6x - 3

E

D G

F33x - 1

115°H

G

E

F

4x - 1

7W

X U

V

2x - 11x

G

F

H

E

Skills Practice State the most specific name for each figure. 1) 2) 3) 4)

Solve for x. Each figure is a parallelogram. 5) 6) 7) 8) 9) 10)

11

Y

FG

EH

V

Z

YX

W

9x + 5

-4 + 10x

K

JM

L

12x + 7

8x - 7U

T

V

W

2x - 11

x + 1

D

CB

A

J

T

U V

W

11) FH = 38, YH = 2x - 1 12) YW = 28, VW = 2x - 8

Find the measurement indicated in each parallelogram. 13) Find m L 14) Find m W 15) Find CB 16) VJ = x + 11, JT = -9 + 3x. Find VJ

12

Skills Review

Formulas:

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =𝑥1+𝑥2

2,

𝑦1+𝑦2

2 𝑆𝑙𝑜𝑝𝑒 =

𝑦2−𝑦1

𝑥2−𝑥1 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

Find the slope between the two points given. Leave your answers in simplified fractional form. 1. (6,-12) and (-16,-13) 2. (9,17) and (-12,7)

3. (6,5) and (20, -10) 4. (17,-7) and (-8,11) 5. (-8, 1) and (6,8) 6. (-7,-16) and (11,-16)

Find the distance between the two given points. Leave your answers in simplified radical form. 7. (0,3) and (-4, 6) 8. (-7,3) 9. (-3,-6) and (-2,1) 10. (4,7) and (-2,-5) 11. (7,-6) and (-4,-7) 12. (0,4) and (9,-2)

13

Find the midpoint for each set of given points. 13. (-10, -5) and (4,-1) 14. (-4, -9) and (-10,-11) 15. (3,-5) and (0,-2) 16. (9,-1) and (10,-11) 17. (-9, -4) and (3,-3) 18. (-12, -3) and (7,9) Find the slope of the lines that would be parallel and perpendicular to the points. 19. (6,-12) and (-16, -13) Slope of parallel line: Slope of perpendicular line: 20. (9,17) and (-12, 7) Slope of parallel line: Slope of perpendicular line: 21. (6,5) and (20, -10) Slope of parallel line: Slope of perpendicular line: 22. (-7,-16) and (11, -16) Slope of parallel line: Slope of perpendicular line:

14

Proving Quadrilaterals on a Coordinate Plane Recall definitions and properties of Special Parallelograms:

Parallelogram- 1. 2. 3. 4.

Rectangle- 1.

Rhombus- 1. 2.

Square- 1. 2.

Guided Practice:

1. Plot points A(-3, -1), B(-1, 2), C(4, 2), and D(2, -1).

a. Find the length of all four sides.

b. Find the slope of all four sides.

c. What specialized geometric figure is quadrilateral ABCD? How do you know?

15

2. Plot points E(1, 2), F(2, 5), G(4, 3) and H(5, 6).

a. Find the length of all four sides.

b. Find the slope of all four sides.

c. What specialized geometric figure is quadrilateral EFHG? How do you know?

d. Describe another way that we could have shown that this figure was a rhombus? 3. Plot points A(1, 0), B(-1, 2), C(2, 5), and D(4,3).

a. Find the length of all four sides.

b. Find the slope of all four sides.

c. What specialized geometric figure is quadrilateral ABCD? How do you know?

d. Describe another way that we could have shown that this figure was a rectangle?

16

Classwork 1. Plot the points W(2, -1), X(1, 3), Y(6, 5), and Z(7, 1).

a. What properties do you need to prove WXYZ is a parallelogram? b. Show that WXYZ is a parallelogram.

2. Plot the points P(3, 1), Q(3, -3), R(-2, -3), and S(-2, 1).

a. What properties do you need to prove PQRS is a rectangle.? b. Show that PQRS is a rectangle.

17

3. Plot the points P(5, 2), Q(1, 9), R(-3, 2), and S(1, -5).

a. What properties do you need to prove PQRS is a rhombus?

b. Show that PQRS is a rhombus. 4. Plot the points P(5, 2), Q(2, 5), R(-1, 2), and S(2, -1).

a. What properties do you need to prove to show PQRS is a square?

b. Show that PQRS is a square.

18

Homework Determine whether the given points represent the vertices of a parallelogram, rectangle, rhombus, or square. Justify your answer mathematically.

1. A(-2, 8), B(5, 8), C(2, 0), D(-5, 0)

2. P(2, 5), Q(-4, 5), R(2, -7), S(-4, -7)

19

3. J(-5, 6), K(-4, -2), L(4, -1), M(3, 7)

4. P(5, 1), Q(9, 6), R(5, 11), S(1, 6)

20

Directions: Question 5 is multiple choice. Choose the correct answer. 5. Three vertices of a rectangle on the coordinate plane are (-2, -1), (6, -1), and (-2, 1).

Which of the following is the coordinate of the fourth vertex? A. (6,1) B. (6, -1) C. (-7, 1) D. (2, 1)

6. Ashanti is surveying for a new parking lot shaped

like a parallelogram. She knows that three of the vertices of parallelogram ABCD are (0,0), (5,2), and (6,5). Find the coordinates of point D.

7. A parallelogram has vertices L(-2, 5), M(3, 3), and N(1, 0). What are possible coordinates for its fourth vertex? Explain.