Geometry Worksheet Quadrilaterals Section: Name:

Geometry Worksheet Quadrilaterals Do Now: Section: Name: 9. The notation of parallelogram: ____________________. Write the definition of quadril...
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Geometry Worksheet

Quadrilaterals

Do Now:

Section:

Name:

9. The notation of parallelogram: ____________________.

Write the definition of quadrilaterals:

A

B

________________________________________________. D

Parts & Properties of the Quadrilaterals

C

Q

P

Name the parallelogram and show its parallel sides:

S

________________________________________________

R

_______________________________________________.

1. ______________________________________________: Vertices that are endpoints of the same side.

Theorems of Parallelogram

Example: ______________________________________.

10. Theorem of Dividing Diagonals A diagonal divides a parallelogram into two congruent triangles

2. ______________________________________________: Sides that have a common endpoint.

Given: ABCD is a parallelogram Prove: ∆ ABD ≅ ∆ CDB Proof:

Example: ______________________________________. 3. ______________________________________________: Sides that do not have a common endpoint.

_______________________________________________.

Example: ______________________________________. 4. ______________________________________________: Angles whose vertices are consecutive. Example: ______________________________________. 5. ______________________________________________: Angles whose vertices are not consecutive. Example: ______________________________________. 6. ______________________________________________: A line segment whose endpoints are two nonadjacent vertices of the quadrilateral.

11. Theorem of Opposite Sides Opposite sides of a parallelogram are congruent Given: ABCD is a parallelogram Prove: BC ≅ DA Proof:

Example: ______________________________________. 7. The sum of the measures of the angles of a quadrilateral is:



_______________________________________________. €

____________________________________________. Example: ______________________________________.

Parallelograms 8. Write the definition of parallelogram: ________________________________________________ ________________________________________________. Mr. Lin

1

Geometry Worksheet

Quadrilaterals

Name: A

x+7

A

Section:

15. ABCD is a parallelogram, ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? 6

DoNow: ABCD is a parallelogram, what’s the perimeter of ABCD ? B

y+2

3

4 O

2y – 6 D

D

B

C

C

3x - 3

16. ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y. A

12. Theorem of Opposite Angles Opposite angles of a parallelogram are congruent

B 2y–6

x+4

∠A ≅ ∠C, and ∠B ≅ ∠D

y+2

D

Given: ABCD is a parallelogram Prove: ∠A ≅ ∠C, and ∠B ≅ ∠D Proof:

O

3x–4 C

________________________________________________

17. Theorem of Consecutive Angles The consecutive angles of a supplementary

parallelogram

are

Given: ABCD is a parallelogram Prove: ∠A and ∠B are supplementary ∠C and ∠D are supplementary ∠A and ∠D are supplementary ∠B and ∠C are supplementary Proof: 13. ABCD is a parallelogram, what are the values of x and y? A B o o x + 20

D

180 o – y

________________________________________________

y – 20

2x–60 o

C

14. Theorem of Bisecting Diagonals The diagonals of a parallelogram bisect each other Given: ABCD is a parallelogram Prove: AC and BD bisect each other at O Proof:



________________________________________________ € 18. ABCD is a parallelogram, what are the values of x, y A and z? o o 120

D

Mr. Lin

zo

x

yo

C

2

B

Geometry Worksheet

Quadrilaterals

19. ABCD is a parallelogram, what are the values of x and y? A B o o x+30

x–30

Name:

23. Given: ABCD is a parallelogram, BO ≅ OD Prove: EO ≅ OF E A



y +20o

D

Section:



C





B

O

D

C F

20. ABCD is a parallelogram, calculate the perimeter of x + 30 ABCD? A B 2y – 10

y + 10

D

C

2x – 10

24. Given: ABCD is a parallelogram, AF || CE Prove: ∠FAB ≅ ∠ECD A

B

E



F

21. ABCD is a parallelogram, solve for x. A

D x+30

C

B

x–10

O

x+10

2x

D

22. Given: ABCD is a parallelogram Prove: XO ≅ YO



C

X

A

B

Review: Theorems of Parallelogram

O





________________________________________________ D

C Y

________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Mr. Lin

3

Geometry Worksheet

Quadrilaterals

Prove Quadrilaterals are Parallelograms 25. List the criteria parallelograms:

for

proving

quadrilaterals

are

(A) _____________________________________________

Section:

Name:

29. Congruent Opposite Sides If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram Given: If AB ≅ CD , and BC ≅ DA Prove: ABCD is a parallelogram Proof: ________________________________________________ € € € €

(B) _____________________________________________ (C) _____________________________________________ (D) _____________________________________________ (E) _____________________________________________ (F) _____________________________________________ 26. Parallel Opposite Sides If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram Given: If AB || CD , and BC || DA Prove: ABCD is a parallelogram Proof: ________________________________________________ € € € €

30. Application Example ABCD is a quadrilateral as shown below, solve for x 5 x + 50 6

6 2x – 30 5

27. Application Example If m∠1 = m∠2 = m∠3, then ABCD is a parallelogram A

1

B 2

31. Application Example ABCD is a parallelogram, if DF = BE, then AECF is also a parallelogram E A

B

3 D

C D

C F

28. Application Example ABCD is a quadrilateral as shown below, solve for x 3x – 20 50o

60o

60o

50o

2x + 10

Mr. Lin

4

Geometry Worksheet

Quadrilaterals

32. Congruent & Parallel Opposite Sides If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram Given: If AB || CD , and AB ≅ CD Prove: ABCD is a parallelogram Proof: ________________________________________________ € € € €

Section:

Name:

35. Congruent Opposite Angles If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Given: ∠A ≅ ∠C, and ∠B ≅ ∠D Prove: ABCD is a parallelogram Proof: ________________________________________________

33. Application Example ABCD is a quadrilateral as shown below, solve for x and y. x+5 y + 50

36. Application Example ABCD is a quadrilateral, solve for x

30o

10

x + 30

10 30o

130o

50o

2y – 20 50o

130o 2x – 40 5

34. Application Example ABCD is a parallelogram, if m∠1 = m∠2, then AECF is also a parallelogram E A

37. Application Example If m∠1 = m∠2 = m∠3, then ABCD is a parallelogram. 1 A

B

4 B

1 3

2 D

D

C

2

C F

Mr. Lin

5

Geometry Worksheet

Quadrilaterals

38. Bisecting Diagonals If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Given: AC and BD bisect each other at O Prove: ABCD is a parallelogram Proof: ________________________________________________ € €

Section:

Name:

41. Application Example ABCD is a quadrilateral, solve for x. 2xo+80o

3xo

100o–2xo

2(xo+45o)–10o o

Review: proving quadrilaterals are parallelograms: (A) ____________________________________________ (B) ____________________________________________ (C) ____________________________________________ (D) ____________________________________________ 39. Application Example ∆ AOB ≅ ∆ COD, then ABCD is a parallelogram

(E) ____________________________________________

A

B

(F) ____________________________________________

O

D

C

40. Supplementary Consecutive Angles If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram Given: ∠A and ∠B are supplementary, and ∠A and ∠D are supplementary Prove: ABCD is a parallelogram Proof: ________________________________________________

Mr. Lin

6

Geometry Worksheet

Quadrilaterals

Section:

Name:

47. Proving Rectangles:

Do Now: 42. List the Properties/Theorems of Parallelograms

(1) _____________________________________________

(1) ______________________________________________ (2) _____________________________________________ (2) ______________________________________________ (3) ______________________________________________ (4) ______________________________________________ (5) ______________________________________________

(3) _____________________________________________ 48. Theorem: If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle Given: ABCD is a parallelogram and m∠A = 90 A Prove: ABCD is a rectangle

Rectangles

D

B

C

43. Write the definition of rectangle: ________________________________________________. 44. Theorem: All angles of a rectangle are right angles Given: ABCD is a rectangle with ∠A = 90o Prove: ∠B = 90 o, ∠C = 90 o, ∠D = 90 o A

D

45. Theorem: The diagonals of a rectangle are congruent Given: ABCD is a rectangle A Prove: AC ≅ BD D



B

49. Theorem: If a quadrilateral is equiangular, it is a rectangle Given: ABCD is a quadrangular & m∠A = m∠B = m∠C A = m∠D Prove: ABCD is a rectangle

C

B

C



46. Properties of Rectangle: (1) _____________________________________________

C

D

50. Theorem: The diagonals of a parallelogram are congruent A Given: AC ≅ BD Prove: ABCD is a rectangle D





51. Application Example: ABCD is a parallelogram, m∠A = 6x - 30 and m∠C = 4x + 10. Show that ABCD is a rectangle

(2) _____________________________________________ (3) _____________________________________________ Mr. Lin

B

7

B

C

Geometry Worksheet

Quadrilaterals

Section:

Name:

Rhombuses

57. Proving Rhombuses:

52. Write the definition of rhombuses:

(1) _____________________________________________

_______________________________________________.

(2) _____________________________________________

53. Theorem: All sides of a rhombus are congruent Given: ABCD is a rhombus with AB ≅ DA A Prove: AB ≅ BC ≅ CD ≅ DA

B

D









58. Theorem: If a parallelogram has two congruent consecutive sides, then the parallelogram is a rhombus Given: ABCD is a parallelogram and AB ≅ DA Prove: ABCD is a rhombus A



54. Theorem: The diagonals of a rhombus are perpendicular to each other A Given: ABCD is a rhombus Prove: AC ⊥ BD D



(4) _____________________________________________

C



(3) _____________________________________________



B



D

C

59. Theorem: If a quadrilateral is equilateral, it is a rhombus Given: ABCD is a parallelogram and AB ≅ BC ≅ CD ≅ DA A Prove: ABCD is a rhombus

C







D€



B

B

C

60. Theorem: If the diagonals of a parallelogram are perpendicular, it is a rhombus B A Given: AC ⊥ BD Prove: ABCD is a rhombus D

€ 55. Theorem: The diagonals of a rhombus bisect its angles A Given: ABCD is a rhombus Prove: AC bisects ∠DAB and ∠DCB BD bisects ∠CDA and ∠CBA D

€ €

B

C

C



61. Theorem: If each diagonal of a parallelogram bisects two opposite angles, then it is a rhombus Given: AC bisects ∠DAB and ∠DCB B A BD bisects ∠CDA and ∠CBA Prove: ABCD is a rhombus

€ €

D

C

62. Application Example: ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, B AD = x + 13 A Prove: ABCD is a rhombus D

C

56. Properties of Rhombuses: (1) _____________________________________________ (2) _____________________________________________

63. Application Example: ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus.

(4) _____________________________________________ Mr. Lin

B

A

(3) _____________________________________________ D

C

8

Geometry Worksheet

Quadrilaterals A

Section:

Name:

B

Squares 64. Write the definition of squares: D _______________________________________________. C _______________________________________________. _______________________________________________. _______________________________________________. 65. Properties of Squares: (1) _____________________________________________ (2) _____________________________________________ (3) _____________________________________________ 66. Proving Squares: (1) _____________________________________________ (2) _____________________________________________ 67. Application Example: ABCD is a square, m∠A = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y.

Mr. Lin

9