Mar 9, 2012 ... Geometry Worksheet. Quadrilaterals ... Parts & Properties of the Quadrilaterals. 1.
... The sum of the measures of the angles of a quadrilateral is: ...
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
