math 2 unit 5 worksheet 1 name: polygon angle-sum theorems

Math 2 Unit 5 Worksheet 1

Math 2 Unit 5 Worksheet 1 Name: Polygon Angle-Sum Theorems Date: Per:

[1-3] Find the sum of the angle measures of each polygon. 1. 2. 3. 12-gon

[4-6] Find the measure of one angle in each regular polygon. Round to the nearest tenth if necessary.

4. 5. 6. regular 15-gon [7-9] Find the missing angle measures.

7. 8. 9. [10-12] Find the measure of an EXTERIOR angle of each REGULAR polygon. Round to the nearest tenth if necessary.

10. Decagon 11. 16-gon 12. hexagon


𝒚𝒚°

𝒙𝒙°

[13-15] Find the value of each variable.

13. 𝑥𝑥 = _________ 14. 𝑥𝑥 = _________ 15. 𝑤𝑤 = _________ 𝑦𝑦 = _________ 𝑦𝑦 = _________ 𝑦𝑦 = _________ 𝑥𝑥 = _________ 𝑧𝑧 = _________

[16-18] The measure of an EXTERIOR angle of a REGULAR polygon is given. Find the measure of an INTERIOR angle. Then find the NUMBER OF SIDES.

16. 120° 17. 18° 18. 12° Interior Angle: _________ Interior Angle: _________ Interior Angle: _________ Number of Sides: _______ Number of Sides: _______ Number of Sides: _______

19. A regular polygon has an interior angle with a measure of 150°. How many sides does the polygon have?

20. A regular polygon has an interior angle with a measure of 175°. How many sides does the polygon have?

21. Find the value of 𝑥𝑥. Drawing is not to scale.

𝑦𝑦° 𝑥𝑥°

103° 93°

87°


Math 2 Unit 5 Worksheet 2 Name: Properties of Parallelograms Date: Per: [1-5] The given figures are all parallelograms. Solve for the variable(s) in each diagram, then state the property of parallelograms that allowed you to solve for the variable(s).

1. 𝑥𝑥 = ____________

Property: _______________________________

_______________________________________

_______________________________________

2. 𝐷𝐷𝐷𝐷 = 2𝑥𝑥 + 2, 𝐷𝐷𝐷𝐷 = 3𝑥𝑥 + 1, 𝐷𝐷𝐸𝐸 = 3𝑥𝑥 − 3, 𝐷𝐷𝐸𝐸 = 2𝑥𝑥 + 6 𝑥𝑥 = ____________

Property: _______________________________

_______________________________________

_______________________________________

3. 𝑎𝑎 = ____________

Property: _______________________________

_______________________________________

_______________________________________

4. 𝑥𝑥 = ____________

𝑦𝑦 = ____________

Property: _______________________________

_______________________________________

_______________________________________

5. 𝑥𝑥 = ____________ 𝑦𝑦 = ____________

Property: _______________________________

_______________________________________

_______________________________________


[6-11] Can you prove the quadrilateral is a parallelogram based on the given information? Explain. 6. 7.

_________________________ _________________________

_________________________ _________________________

_________________________ _________________________

8. 9. _________________________ _________________________ _________________________ _________________________ _________________________ _________________________

10. 11.

_________________________ _________________________ _________________________ _________________________ _________________________ _________________________

[12-15] For what values of 𝑥𝑥 and 𝑦𝑦 must each figure be a parallelogram?

12. 𝑥𝑥 =________ 13. 𝑥𝑥 =________

𝑦𝑦 =________ 𝑦𝑦 =________

14. 𝑥𝑥 =________ 15. 𝑥𝑥 =________

𝑦𝑦 =________ 𝑦𝑦 =________

16. A classmate draws a parallelogram for which one side is twice as long as the other. If one side is 26 units, what are all the possible lengths of the perimeter?

17. Complete the two-column proof.

Given: Parallelogram 𝐴𝐴𝐴𝐴𝐴𝐴𝐷𝐷 with 𝐴𝐴𝐴𝐴�� ≅ 𝐴𝐴𝐷𝐷�� Prove: ∆𝐴𝐴𝐷𝐷𝐴𝐴 ≅ ∆𝐴𝐴𝐴𝐴𝐷𝐷

Statement Reason

1. Parallelogram 𝐴𝐴𝐴𝐴𝐴𝐴𝐷𝐷 with 𝐴𝐴𝐴𝐴 ≅ 𝐴𝐴𝐷𝐷

1. Given

2.

2. Opposite sides of parallelograms are congruent.

3. 𝐷𝐷𝐴𝐴�� ≅ 𝐴𝐴𝐷𝐷�� 4.

3. 4. SSS


5. List 3 things that are

TRUE about the

diagonals of a rhombus.

10. List 2 things that are

TRUE about the

diagonals of a rectangle.

Math 2 Unit 5 Worksheet 3 Name: Properties of Rhombuses, Rectangles, and Squares Date: Per:

[1-4] Decide whether the parallelogram is a rhombus, a rectangle, or a square. Explain. 1. 2.

_________________________ _________________________

_________________________ _________________________

_________________________ _________________________

3. 4. _________________________ _________________________ _________________________ _________________________ _________________________ _________________________

1. ____________________________________________________________________ 2. ____________________________________________________________________ 3. ____________________________________________________________________

[6-9] Find the measures of the numbered angles in each rhombus.

6. 7. 𝑚𝑚∠1 ______ 𝑚𝑚∠2 ______ 𝑚𝑚∠1 ______ 𝑚𝑚∠2 ______

𝑚𝑚∠3 ______ 𝑚𝑚∠4 ______ 𝑚𝑚∠3 ______ 𝑚𝑚∠4 ______

8. 9. 𝑚𝑚∠1 ______ 𝑚𝑚∠2 ______ 𝑚𝑚∠1 ______ 𝑚𝑚∠2 ______ 𝑚𝑚∠3 ______ 𝑚𝑚∠4 ______ 𝑚𝑚∠3 ______ 𝑚𝑚∠4 ______

1. ____________________________________________________________________ 2. ____________________________________________________________________

[11-12] 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 is a rectangle. Find the value of 𝑥𝑥 and the length of each diagonal.

11. 𝐻𝐻𝐻𝐻 = 3𝑥𝑥 + 5 and 𝐻𝐻𝐻𝐻 = 5𝑥𝑥 − 9 12. 𝐻𝐻𝐻𝐻 = 3𝑥𝑥 + 7 and 𝐻𝐻𝐻𝐻 = 6𝑥𝑥 − 11

𝑥𝑥 = _______ 𝑥𝑥 = _______

𝐻𝐻𝐻𝐻 = ______ 𝐻𝐻𝐻𝐻 = ______

𝐻𝐻𝐻𝐻 = ______ 𝐻𝐻𝐻𝐻 = ______


[13-14] Classify each of the following parallelograms as a rhombus, a rectangle, or a square. Explain.

13. 𝑀𝑀𝑀𝑀�� ≅ 𝑃𝑃𝑃𝑃�� 14. 𝐴𝐴𝐴𝐴�� ≅ 𝐵𝐵𝐵𝐵�� _________________________ _________________________

_________________________ _________________________

_________________________ _________________________

15. For what value of 𝑥𝑥 is parallelogram 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 a rhombus? 16. 𝑆𝑆𝑆𝑆 = 14. For what value of 𝑥𝑥 is parallelogram 𝑃𝑃𝑆𝑆𝑃𝑃𝑆𝑆 a rectangle? Find 𝑃𝑃𝑃𝑃 and 𝑃𝑃𝑃𝑃.

𝑥𝑥 = ______ 𝑥𝑥 = ______ 𝑃𝑃𝑃𝑃 = _____ 𝑃𝑃𝑃𝑃 = _____

[17-24] For what value of 𝑥𝑥 is the figure the given special parallelogram?

17. Rhombus 𝑥𝑥 = _______ 18. Square 𝑥𝑥 = _______

19. Rectangle 𝑥𝑥 = _______ 20. Rhombus 𝑥𝑥 = _______

21. Rhombus 𝑥𝑥 = _______ 22. Rectangle 𝑥𝑥 = _______

23. Rhombus 𝑥𝑥 = _______ 24. Rectangle 𝑥𝑥 = _______


Math 2 Unit 5 Worksheet 4 Name: Properties of Trapezoids Date: Per:

Trapezoids [1-2] The given figures are trapezoids. Solve for the variable(s) in each diagram.

1. 𝑥𝑥 = _______

𝑦𝑦 = _______

2. 𝑥𝑥 = _______

[3-7] Find 𝑥𝑥 and the length of the given segment in each trapezoid.

3. 𝑥𝑥 = _________

𝐺𝐺𝐺𝐺�� = _______

4. 𝑥𝑥 = _________

𝐹𝐹𝐹𝐹�� = _______

5. 𝑥𝑥 = _________ 𝐶𝐶𝐶𝐶�� = _______

6. 𝑥𝑥 = _________ 𝑇𝑇𝑇𝑇�� = _______

7. 𝑥𝑥 = _________ 𝑊𝑊𝑊𝑊�� = _______

(2𝑥𝑥)°

136° (2𝑦𝑦 + 100)°

(4𝑦𝑦 − 82)°

𝑥𝑥°

88°

𝐶𝐶 𝐶𝐶

2𝑥𝑥

10

3𝑥𝑥 − 4

𝐺𝐺 𝐺𝐺

𝐹𝐹 𝐹𝐹

𝐶𝐶 𝐶𝐶

4𝑥𝑥 + 1

10

6𝑥𝑥 − 2

𝐺𝐺 𝐺𝐺

𝐹𝐹 𝐹𝐹

2𝑥𝑥 − 1

3𝑥𝑥 − 6

25

𝐶𝐶 𝐶𝐶

𝐺𝐺

𝐹𝐹

𝐺𝐺

𝐹𝐹


Isosceles Trapezoids 8. What is true about an isosceles trapezoid that is not necessarily true about all trapezoids?

1. _____________________________________________________________________________________

2. _____________________________________________________________________________________

3. _____________________________________________________________________________________ [9-10] Find the measures of the numbered angles in each isosceles trapezoid.

9. 𝑚𝑚∠1= _______ 10. 𝑚𝑚∠1= _______

𝑚𝑚∠2= _______ 𝑚𝑚∠2= _______

𝑚𝑚∠3= _______ 𝑚𝑚∠3= _______

[11-12] Find the value of the 𝑛𝑛 in each isosceles trapezoid.

11. 𝑛𝑛 = _______ 12. 𝑛𝑛 = _______

[13-12] Find the value(s) of the variable(s) in each isosceles trapezoid.

13. 𝑥𝑥 = _______ 14. 𝑥𝑥 = _______

𝑦𝑦 = _______

15. 𝑥𝑥 = _______ 16. 𝐴𝐴𝐶𝐶 = 𝑥𝑥 + 5 𝑥𝑥 = _______ 𝐵𝐵𝐶𝐶 = 2𝑥𝑥 − 2

17. Error Analysis. What is the error in the following reasoning? An iscoceles trapezoid has one pair of parallel sides and one pair of congruent sides. Therefore, an isoceles trapezoid is a parallelogram.

(5𝑛𝑛 + 3)° 48°

𝐴𝐴

𝐵𝐵

𝐶𝐶

𝐶𝐶


18. Is it possible to draw an isoceles trapezoid with one pair of opposite angles congruent? Explain and include a drawing of your explanation.

[19-24] True/False. If false, draw a counter example.

19. True / False: Diagonals of a rhombus must be congruent.

20. True / False: All angles of a rectangle are congruent.

21. True / False: All sides of a rectangle are congruent.

22. True / False: The diagonals of a square form four right triangles.

23. True / False: A trapezoid cannot have a right angle.

24. True / False: The diagonals of a parallelogram biscet the angles.


Math 2 Unit 5 Worksheet 5 Name: Mid-Unit Review Date: Per:

[1-6] Solve for the variables. Give the best name for each of the following based upon given information and calculations. {Names of Quadrilaterals are: Quadrilateral, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, & Isosceles Trapezoid}

1. 1. 𝑥𝑥 = ________ 𝑦𝑦 = ________ Name: ____________________

2. 2. 𝑥𝑥 = ________ 𝑦𝑦 = ________ Name: ____________________

3. 3. 𝑥𝑥 = ________

𝑦𝑦 = ________ 𝑧𝑧 = ________ Name: ____________________

4. 4. 𝑥𝑥 = ________

𝑦𝑦 = ________ 𝑛𝑛 = ________ Name: ____________________

5. 5. 𝑥𝑥 = ________ 𝑦𝑦 = ________ 𝑛𝑛 = ________ Name: ____________________

70°

5𝑥𝑥° 4𝑦𝑦°

𝑦𝑦°

𝑥𝑥°

37°

𝑦𝑦°

𝑥𝑥°

3𝑧𝑧 + 3

8𝑧𝑧 − 7

𝑥𝑥°

𝑦𝑦° 52° (5𝑛𝑛 + 1)

(8𝑛𝑛 − 11)

3𝑛𝑛 + 2

(7𝑥𝑥 − 19)°

24°

5𝑛𝑛 + 7

(5𝑥𝑥 + 3)°

𝑦𝑦°

Perimeter of Quadrilateral is 90 cm


6. 6. 𝑥𝑥 = ________ 𝑎𝑎 = ________ Name: ____________________

7. Find the sum of the measures of the interior angles for the following convex polygon: 17-gon 7. _______________________

8. Find the measure of each interior angle for the following regular polygon: Decagon 8. _______________________

9. Find the measure of each interior angle for the following regular polygon: Pentagon 9. _______________________

10. Find the number of sides for a convex polygon whose interior angle sum is: 3060° 10. _______________________

11. Find the number of sides for the following regular polygons, given: The measure of each exterior angle is 7.5° 11. _______________________

𝐵𝐵 𝐶𝐶

𝐷𝐷 𝐴𝐴

2𝑥𝑥 − 2

3𝑥𝑥 − 3

2𝑥𝑥 + 1

(4𝑎𝑎 + 1)° (6𝑎𝑎 − 13)°


12. Complete the below chart on the Properties of Quadrilaterals

[13-24] Write all, some, or no, then explain.

13. _______________ rectangles are squares. ___________________________________________________________________________________________

14. _______________ isosceles trapezoids are parallelograms.

___________________________________________________________________________________________

15. _______________ rhombuses are quadrilaterals.

___________________________________________________________________________________________

Property Parallelogram Rectangle Rhombus Square Trapezoid Sum of the interior angles is 360°

Two pairs of opposite sides are parallel.

Has exactly one pair of parallel sides.

Two pairs of opposite sides are congruent.

All sides are congruent.

Diagonals are congruent.

Diagonals are perpendicular.

A diagonal bisects two angles.

A diagonal forms two congruent triangles.

Diagonals bisect each other.

Opposite angles are congruent.

All angles are right angles.

Consecutive interior angles are supplementary.


16. _______________ squares are triangles.

___________________________________________________________________________________________

17. _______________ rectangles are regular quadrilaterals. ___________________________________________________________________________________________

18. _______________ quadrilaterals have four congruent angles.

___________________________________________________________________________________________

19. _______________ rectangles are rhombuses.

___________________________________________________________________________________________

20. _______________ trapezoids have one pair of opposite sides parallel.

___________________________________________________________________________________________

21. _______________ trapezoids have two pairs of congruent sides.

___________________________________________________________________________________________

22. _______________ squares are regular quadrilaterals.

___________________________________________________________________________________________

23. _______________ trapezoids have four congruent sides.

___________________________________________________________________________________________

24. _______________ parallelograms have four congruent angles.

___________________________________________________________________________________________


Math 2 Unit 5 Worksheet 6 Name: Using Coordinates to Classify Quadrilaterals Date: Per:

Tools to Classify when Graphing

Slope Formula A. Same Slope = Parallel B. Slopes Opposite Reciprocals = Perpendicular Midpoint Formula C. Same Midpoint = Bisect Each other Distance Formula D. Same length = Congruent

Quadrilaterals and Useful Properties

In each blank, write A, B, C, or D stating which formula should be used to verify each property.

Parallelogram ______Opposite Sides are Parallel ______Opposite Sides are Congruent ______Diagonals Bisect Each other ______One pair opposite sides congruent & parallel Rhombus (A parallelogram with…) ______All sides congruent ______Diagonals Perpendicular Rectangle (A parallelogram with…) ______Adjacent Sides perpendicular ______Diagonals Congruent Square (A parallelogram with…) A property from Rhombus and rectangle Trapezoid ______One pair opposite sides parallel (not both) Isosceles Trapezoid (A trapezoid with…) ______Legs Congruent ______Diagonals Congruent

Graph Work/Justification/Answer

1.

Best Classification: _____________________


2.


3.


4.



5.


6.


7.



8.


9.



Math 2 Unit 5 Worksheet 7 Name: Using Coordinate Geometry Date: Per: to Prove Properties of Quadrilaterals

[1-3] What are the coordinates of the vertices of each figure? 1. Rectangle with 2. Rectangle centered at the 3. Square with height 𝑥𝑥

base 𝑏𝑏 and height ℎ origin with base 2𝑏𝑏 and height 2ℎ

𝐴𝐴 ( ___, ___ ) 𝐵𝐵 ( ___, ___ ) 𝐷𝐷 ( ___, ___ ) 𝐸𝐸 ( ___, ___ ) 𝐻𝐻 ( ___, ___ ) 𝐼𝐼 ( ___, ___ ) 𝐶𝐶 ( ___, ___ ) 𝐷𝐷 ( ___, ___ ) 𝐹𝐹 ( ___, ___ ) 𝐺𝐺 ( ___, ___ ) 𝐽𝐽 ( ___, ___ ) 𝐾𝐾 ( ___, ___ )

[4-12] Determine the missing coordinates in the diagrams. Do not introduce any new variables.

4. 5. 6.

7. 8. 9.

10. 11. 12.

x

y

(𝒅𝒅,𝟎𝟎) (–𝒅𝒅,𝟎𝟎)

( ___,𝒉𝒉) ( ___, ___ )

y

(𝟎𝟎,𝒑𝒑)

(𝒂𝒂,𝟎𝟎)

( ___, ___ )

x

y

(𝒅𝒅, 𝒓𝒓)

( ___, ___ )

( ___, ___ )

(𝒔𝒔,𝟎𝟎) x

y

(𝟎𝟎, – 𝒓𝒓 ) ( ___, ___ )

x

y

(𝒔𝒔 + 𝒃𝒃,𝒘𝒘) (𝒃𝒃,𝒘𝒘)

y

(𝟎𝟎,𝒂𝒂 )

(𝟐𝟐𝒂𝒂,𝟎𝟎 )

(–𝒌𝒌,𝟎𝟎 ) x

y

(𝒌𝒌, – 𝒄𝒄 ) ( ___, ___ )

( ___, ___ ) ( ___, ___ ) x

y

(–𝒂𝒂, –𝒃𝒃) ( ___, ___ )

x

y

(𝒈𝒈,𝟎𝟎)

(𝒄𝒄,𝒌𝒌) ( ___, ___ )

( ___, ___ )

( ___, ___ )


13. Find the missing coordinates and find the midpoint of each diagonal.

14. Quadrilateral FLEA is a parallelogram. Find the coordinates for points F and E.

a) Show opposite sides are parallel using ____________________________ (which formula)

b) Show opposite sides are congruent using ____________________________ (which formula)

c) Show diagonals bisect each other using ____________________________ (which formula)

( ___, ___ ) ( 𝒓𝒓, 𝒔𝒔 )

( 𝒕𝒕,𝟎𝟎 ) ( ___, ___ )

𝑭𝑭 (______, ______)

𝑬𝑬 (______, ______)

x

y

𝑳𝑳 (𝒏𝒏 + 𝒎𝒎,𝒑𝒑)

𝑨𝑨(−𝒏𝒏,𝟎𝟎)

𝑭𝑭

𝑬𝑬

Math 2 Unit 5 Review Worksheet

Math 2 Unit 5 Name: Review Worksheet Date: Per:

[1-20] Show all work for each problem. NOTE: Diagrams are not drawn to scale.

1. What is the value of 𝑥𝑥? 2. Determine the value of 𝑥𝑥. A. 540° A. 15

B. 390° B. 15.4 C. 150° C. 9 D. 120° D. 19.8

3. Determine the sum of the exterior angles of an octagon. A. 1440°

B. 1080° C. 360°

D. 135° 4. Determine the measure of each interior angle of a regular sided polygon with 9 sides. A. 1620°

B. 180° C. 1260°

D. 140° 5. Determine the measure of each exterior angle of a regular polygon with 12 sides. A. 30°

B. 150° C. 216°

D. 36° 6. The measure of an interior angle of a regular polygon is 162°. How many sides does the polygon have? A. 18 sides

B. 20 sides C. 16 sides

D. 10 sides 7. Determine the value of 𝑥𝑥? A. 80° B. 40° C. 60° D. 20°

120°

𝑥𝑥°

100°

100° 140°

140°

100°

𝑥𝑥°

(8𝑥𝑥 + 1)°

(5𝑥𝑥 – 4)° 73°

95°


8. Find the number of sides for a regular polygon if each exterior angle has a measure of 15°. A 2340

B. 180 C. 24

D. 26 9. If 𝐹𝐹𝐹𝐹 = 30, find 𝐹𝐹𝐹𝐹. A. 12 B. 18 C. 15 D. 30

10. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 is a rhombus. Determine the value of 𝑥𝑥. A. 110°

B. 55° C. 70° D. 35°

11. What are the values of the variables in the given parallelogram?

A. 𝑥𝑥 = 7, 𝑦𝑦 = 9 B. 𝑥𝑥 = 7, 𝑦𝑦 = 65 C. 𝑥𝑥 = 5, 𝑦𝑦 = 71 D. 𝑥𝑥 = 3, 𝑦𝑦 = 77

Rule/Property used to solve for: 𝑥𝑥 𝑦𝑦

12. If 𝐹𝐹𝐾𝐾𝐾𝐾𝐾𝐾 is a rhombus, and 𝑚𝑚∠𝐹𝐹𝐾𝐾𝐾𝐾 = 70°, what is the measure of 𝑚𝑚∠1? A. 55°

B. 50° C. 35° D. 90° 13. Which statements are true for a parallelogram? Select all that apply.

A. 𝑚𝑚∠𝐴𝐴 + 𝑚𝑚∠𝐵𝐵 = 180° B. 𝑚𝑚∠𝐴𝐴 + 𝑚𝑚∠𝐶𝐶 = 180°

C. ∠𝐴𝐴 ≅ ∠𝐷𝐷 D. 𝐵𝐵𝐶𝐶�� ≅ 𝐴𝐴𝐷𝐷��

E. ∠𝐴𝐴 ≅ ∠𝐶𝐶

12

F

J H

G

K

A

B C

D

N M

L K 1

Q P

V T

110° 𝑥𝑥°

(2𝑦𝑦 + 16)° (6𝑥𝑥 – 8)°

(4𝑥𝑥 + 6)°


14. Determine the length of 𝐹𝐹𝐾𝐾�� in the trapezoid shown. A. 26 B. 4 C. 13 D. 17 15. a) If 𝑚𝑚∠𝐴𝐴 is 80°, find the measures of:

𝑚𝑚∠𝐵𝐵 = ______

𝑚𝑚∠𝐶𝐶 = ______

𝑚𝑚∠𝐷𝐷 = ______

b) What type of quadrilateral is 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷? Be as specific as possible. _________________________________________________ 16. Determine the values of x and y. 𝑥𝑥 = _________ 𝑦𝑦 = _________

17. 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷 is an isosceles trapezoid with midsegment 𝐸𝐸𝐹𝐹��. Determine the following:

𝑛𝑛 = _____________ 𝐸𝐸𝐹𝐹 = ______________ 𝑥𝑥 = ______________ 𝑚𝑚∠𝐴𝐴𝐷𝐷𝐶𝐶 = _________

Rules/Properties used to solve for: 𝑥𝑥 𝑛𝑛

M N

L K

I H

4𝑥𝑥 + 1

5𝑥𝑥 + 2

27

(2𝑥𝑥 – 17)° A

C B

E

10

D

F 3𝑛𝑛 – 4

36

(3𝑥𝑥 + 2)°

52°

𝑦𝑦°

𝑥𝑥°

Rules/Properties used to solve for 𝑥𝑥:

Rule/Property used to solve for 𝑥𝑥:

4

10 A D

C B

6 6


18. 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷 is a parallelogram. Determine the following: 𝑥𝑥 = _____________

𝑦𝑦 = _____________

19. 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷 is a rhombus. Determine the following: 𝑥𝑥 = _____________ 𝑦𝑦 = _____________ 𝑛𝑛 = _____________ 𝑧𝑧 = _____________ 20. For rectangle 𝑆𝑆𝑃𝑃𝑃𝑃𝑆𝑆 find the coordinates of 𝑃𝑃 (without using any new variables), and

find the midpoints of 𝑃𝑃𝑆𝑆�� and 𝑆𝑆𝑃𝑃��.

𝑃𝑃 ( ______ , ______ ) Midpoint of 𝑃𝑃𝑆𝑆�� ( ______ , ______ )

Midpoint of 𝑆𝑆𝑃𝑃�� ( ______ , ______ ) 21. For parallelogram 𝐴𝐴𝐴𝐴𝐾𝐾𝐾𝐾 find the coordinates of 𝐴𝐴 and 𝐾𝐾 (without using any new variables), and

find the midpoints of 𝐴𝐴𝐾𝐾�� and 𝐾𝐾𝐴𝐴�� 𝐴𝐴 ( ______ , ______ ) 𝐾𝐾 ( ______ , ______ )

Midpoint of 𝐴𝐴𝐾𝐾�� ( ______ , ______ )

Midpoint of 𝐾𝐾𝐴𝐴�� ( ______ , ______ )

Rule/Property used to solve: 𝑦𝑦

𝑛𝑛

42

B

A D

C

E 2𝑥𝑥 + 9𝑦𝑦

15 – 3𝑦𝑦

𝑥𝑥

𝑃𝑃 (0,𝑘𝑘)

y

x 𝑆𝑆 (𝑐𝑐, 0)

𝑃𝑃 ( ____ , ____ )

𝑆𝑆 (0, 0)

𝑥𝑥° D

C B

A

𝑦𝑦°

15𝑛𝑛 – 7

43°

12𝑛𝑛 + 20

𝑧𝑧°

Rule/Property used to solve for 𝑥𝑥 and 𝑦𝑦:

x

y

𝐴𝐴 (𝑎𝑎 + 𝑏𝑏, 𝑐𝑐)

𝐾𝐾(−𝑎𝑎, 0) 𝐾𝐾( _____ , _____ )

𝐴𝐴( _____ , _____ )


22. Plot the points 𝐴𝐴(1, – 1), 𝐵𝐵(2, – 4) and 𝐶𝐶(4, – 2)

A. Plot a fourth point 𝐷𝐷 in B. Plot a fourth point 𝐷𝐷 in C. Plot a fourth point 𝐷𝐷 in quadrant 4 that will make quadrant 3 that will make quadrant 1 that will a parallelogram. a parallelogram. make a parallelogram. 23. Determine whether each statement is ALWAYS, SOMETIMES, or NEVER true

Always Sometimes Never

A rectangle is a square.

A square is a rhombus.

A trapezoid is a parallelogram.

The diagonals of a parallelogram are perpendicular.

The sides of a parallelogram are congruent.

Explain why the quadrilateral formed is a parallelogram using the slope formula.

Explain why the quadrilateral formed is a parallelogram using the midpoint formula.

Explain why the quadrilateral formed is a parallelogram using the distance formula.


24. Samantha must prove this theorem: If 𝐴𝐴𝐵𝐵�� ≅ 𝐷𝐷𝐶𝐶�� and 𝐵𝐵𝐶𝐶�� ≅ 𝐷𝐷𝐴𝐴��, then 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷 is a parallelogram.

Which choice correctly fills in the blank line in the paragraph proof?

A. SAS

B. ASA

C. AAS

D. SSS

E. HL

F. CPCTC

𝐴𝐴𝐵𝐵�� ≅ 𝐶𝐶𝐷𝐷�� and 𝐵𝐵𝐶𝐶�� ≅ 𝐷𝐷𝐴𝐴�� because of given information. 𝐴𝐴𝐶𝐶�� ≅ 𝐴𝐴𝐶𝐶�� by the reflexive property.

∆𝐴𝐴𝐵𝐵𝐶𝐶 ≅ ∆𝐶𝐶𝐷𝐷𝐴𝐴 by the__________ theorem. ∠1 ≅ ∠4 and ∠2 ≅ ∠3 because corresponding parts of

congruent triangles are congruent. 𝐴𝐴𝐵𝐵�� ∥ 𝐷𝐷𝐶𝐶�� and 𝐵𝐵𝐶𝐶�� ∥ 𝐴𝐴𝐷𝐷�� because if alternate interior angles are

congruent then the lines are parallel. 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷 is a parallelogram by definition of parallelogram.

A

B C

D 2

3 4

1

math 2 unit 5 worksheet 1 name: polygon angle-sum theorems

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