quadrilaterals - Lincoln-Sudbury Regional High School S2/Quads... · Quadrilaterals Tree of quadrilaterals Quadrilateral ... has all properties of a quadrilateral and ... K. May a

Quadrilaterals

Tree of quadrilaterals

Quadrilateral

Parallelogram

Trapezoid

Rectangle Rhombus

IsoscelesTrapezoid

Square

Quadrilaterals page 1

Properties of quadrilaterals (in writing)

Quadrilateral1)

Parallelogram: has all properties of a quadrilateral and....1)

2)

3)

4)

5)

Rectangle: has all properties of a parallelogram and.....

1)

2)

Rhombus: has all properties of a parallelogram and....

1)

2)

3)

Square: has all properties of a parallelogram and a rhombus

Trapezoid: has all properties of a quadrilateral and......

1)

2)

Isosceles Trapezoid: has all properties of a trapezoid and.....

1)

2)

3)


Properties of quadrilaterals (with marked pictures)

Quadrilateral

Parallelogram: has all properties of a quadrilateral and....

Rectangle: has all properties of a parallelogram and.....

Rhombus: has all properties of a parallelogram and....

Square: has all properties of a parallelogram and a rhombus

Trapezoid: has all properties of a quadrilateral and......

Isosceles Trapezoid: has all properties of a trapezoid and.....


Problems:

1. Name the quadrilateral which make the statement true.

A. In every ______________, _____________, ____________, and _____________ the diagonals bisect each other.

B. In every ______________ , ______________ , and______________ the diagonals are equal.

C. In every______________ the diagonals are equal and perpendicular.

D. In every ______________ and ______________the diagonals are perpendicular.

2. In a rectangle,

A. The opposite sides are both _____________ and _____________.

B. The diagonals are ______________ to each other but are not necessarily

_____________________ to each other.

C. The diagonals ________________ each other.

D. The angles are all ______________ angles.

E. A rectangle is also a ____________________.

3. In a rhombus,

A. All four sides are ______________ and the opposite sides are ______________.

B. The diagonals are ____________________ to each other and they

______________ each other, but they are not always _________ to each other.

C. A rhombus is also a ___________________.


4. In a square,

A. All four sides are ______________and the opposite sides are ______________.

B. All four angles are ______________ angles.

C. The diagonals are both______________ and ________________ to each other.

D. The diagonals also ______________ each other.

E. A square is also a ____________, a _____________ and a ____________.

5. Answer yes or no:

A. Must the diagonals of a rectangle be perpendicular? _______

B. Are the opposite angles of a rhombus always equal? _______

C. Are all squares rhombuses? ________

D. Are some trapezoids parallelograms? ______

E. May a rectangle have four equal sides? ______

F. Are there some rectangles that are also rhombuses? _____

G. Does the diagonal of a rectangle divide it into two congruent triangles? _____

H. Must a quadrilateral with four equal sides be a square? ______

I. Must a parallelogram with a right angle be a rectangle? ______

J. May a trapezoid have two right angles? ______

K. May a trapezoid have three right angles? ______

L. Must the diagonals of a rhombus be equal? _______

M. May a trapezoid have three equal sides? _______

N. Must the diagonals of a rhombus be perpendicular? ______


Answer “True” or “False”:

____ 1. A trapezoid can be a parallelogram.

____ 2. No rectangle is a rhombus.

____ 3. A square is a rectangle.

____ 4. A parallelogram is a quadrilateral.

____ 5. An isosceles trapezoid is a parallelogram.

____ 6. All rhombuses are squares.

____ 7. All squares are rhombuses.

____ 8. No trapezoid is a rectangle.

____ 9. A rhombus is a parallelogram.

Answer “Yes” or “No”:

____ 10. Must the diagonals of a rectangle be perpendicular?

____ 11. Are the opposite angles of a rhombus always equal?

____ 12. Are all squares rhombuses?

____ 13. Are some trapezoids parallelograms?

____ 14. May a rectangle have four equal sides?

____ 15. Are there some rectangles that are also rhombuses?

____ 16. Does the diagonal of a rectangle divide it into two congruent triangles?

____ 17. Must a quadrilateral with four equal sides be a square?

____ 18. Must a parallelogram with a right angle be a rectangle?

____ 19. May a trapezoid have two right angles?

____ 20. May a trapezoid have three right angles?

____ 21. Must the diagonals of a rhombus be equal?


____ 22. May a trapezoid have three equal sides?

____ 23. Must the diagonals of a rhombus be perpendicular?

Answer always sometimes never for the following:

_____24. A trapezoid has 3 right angles

_____25. A trapezoid has 2 right angles

_____26. A parallelogram has a exactly one right angle

_____27. A parallelogram has at least one right angle

_____28. The opposite angles of a quadrilateral are equal

_____29. The opposite angles of a rhombus are equal

_____30. The opposite angles of a parallelogram are supplementary

_____31. A rectangle is a trapezoid

_____32. A square is a rhombus

_____33. A quadrilateral is a trapezoid

_____34. A parallelogram has exactly one set of parallel sides

_____35. Consecutive sides of a parallelogram are equal

_____36. A rhombus has a 30º angle

_____37. The angles of a trapezoid add up to 360º

_____38. A trapezoid has 3 equal sides

_____39. A trapezoid has 4 equal sides

_____40. A trapezoid has one set of opposite, parallel, equal sides.

_____41 The sides of a quadrilateral add up to 360

_____42. The diagonals of a square are equal

_____43. The diagonals of a parallelogram are equal


____ 44. If one of the angles of a parallelogram is a right angle, then all of the other three angles must be right angles.

____45. The diagonals of a parallelogram are congruent.

____46. A rhombus is a trapezoid.

____47. The diagonals of a square divide the square into four congruent right isosceles triangles.

____48. If the diagonals of a quadrilateral are congruent, the quad is an isosceles trapezoid.

____49. If the diagonals of a quadrilateral divide the vertex angles into two 45o angles,

the quad is a square.

____50. If a parallelogram is equilateral, it is equiangular.

____51. If a parallelogram is equiangular, then it is a rectangle.

____52. If two of the angles of a trapezoid are congruent, then the trapezoid is isosceles.

____53. If the diagonals of a quad are perpendicular bisectors of each other, then the quad is a square.

____54. If a quad. is a parallelogram, then its diagonals are perpendicular bisectors of each other.

____55. If one of a quadrilateral’s diagonals divides the quad into two congruent right triangles, then the quad. is a rectangle.

____56. The diagonals of a rhombus divide it into 4 congruent right triangles.

____57. The diagonals of a rhombus are congruent.


Multiple Choice:

1. Which of the following are equiangular quadrilaterals? a. parallelogram and rectangle b. trapezoid and rhombus c. square and rhombus d. square and rectangle

2. Which of the following is not a property of a parallelogram? a. both pairs of opposite sides are parallel b. both pairs of opposite angles are congruent c. both pairs of opposite sides are congruent d. one pair of opposite sides are congruent and the other pair of opposite sides are parallel e. the diagonals bisect each other

3. Which of the following statements is false? a. a parallelogram is a quadrilateral b. a trapezoid is a parallelogram c. a rectangle is a parallelogram d. a square is a rhombus

4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is always: a. a rhombus b. a rectangle c. a parallelogram

5. The opposite angles of a parallelogram are always: a. equal b. supplementary c. complementary

6. The diagonals of a rectangle are always: a. equal to each other b. equal and perpendicular to each other c. perpendicular and bisect each other


More problems:

1. Given quadrilateral EFGH.

A. Find x

B. Is EFGH a parallelogram? Why?

2. The figure below is a parallelogram. Find x and the perimeter.

14 14

x + 22

4x + 13

x = ___________

Perimeter: ____________


3. Find all angles in trapezoid HKRT.

m∠H = m∠K =

m∠T = m∠R =

4. Which of the following statements is false?

A. A parallelogram is a quadrilateral. B. A rectangle is a square. C. A rectangle is a parallelogram.

5. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is always:

A. a rhombus B. a rectangle C. a parallelogram.

6. The opposite angles of a parallelogram are always:

A. equal B. supplementary C. complementary

7. The diagonals of a rectangle are always:

A. equal to each other B. equal and perpendicular to each other C. perpendicular and bisect each other

8. If one angle of a trapezoid is a right angle, then the greatest number of additional right angles that the trapezoid may have is:

A. 1 B. 2 C. 3

H

T R

K

5x+15

4x-4

85°


9. Find x:

102°

x

84°

75°

10. PQRS is a parallelogram. Find x and then find each of the angles P, Q, R and S.

P

S

5x+1

3x+9R

Q

11.ABCD is an isosceles trapezoid.

∠C = x + 60( )o

∠A = 4x( )o

AD = x − 3( )

Find the length of AD .D

CB

A


13.Which statement is not always true? a. The diagonals of a rectangle are congruent. b. The diagonals of a rhombus are perpendicular bisectors of each other. c. The diagonals of an isosceles trapezoid are congruent. d. The diagonals of a parallelogram bisect the angles of the parallelogram.

14. Given: parallelogram ABCD

m∠B = 5x 2 +10x − 8

m∠D = 4x 2 + 6x + 4

Solve for both values of x, m∠B, and m∠C.

15. ABCD is a rhombus. DB=6 and AC=10. Find the perimeter.

16. ABCD is a rectangle. BC=4 and AB=6. Find the length of BD and AC.

19. QUAD is a parallelogram. Give the most descriptive name for QUAD. UF = 2x–y, FD = 18, QF = x+y, FA = 6+2y,

�

∠QFD = (10x)o

D

CB

A

F

10x( )°

D

AU

Q


20. KMOP is a parallelogram. ∠M = (x + 3y)º , ∠O = (x − 4)º , and ∠P = (4y − 8)º . Find

m∠K .

21. Trapezoid ABCD has ∠C = 70oand ∠D = 50o . Find the measures of ∠A and ∠B .

22. PQRS is a square. QR = 8, PR = PX, and M is the midpoint of PQ . Find MX.

S

X

R

QMP

23. ABCD is a parallelogram. AD =x 2

6+18

and BC =x3

. Solve for x


24. ABCD is a rectangle. AB = 4 + 2 , BC = 4 − 2 , and AC = x . Solve for x.

25. ABCD is a rhombus. AB = 12 and BE = 5. Find AC.

26. Find the dimensions of a rectangular lot if its length is 7 meters greater than its width and each diagonal is 13 m long.

27. ABCD is a rectangle. AB = x , AD = x + 3, and BD = 15 . Find x.

28. In the rectangle below, x, y, and z are consecutive even integers. Set up and solve an algebraic equation to solve for x, y, and z.

Y

XZ

E

A B

CD


29. ABCD is a rectangle. AC = x2 +1 and BD = 4x +1 . Solve for x and the length of AC. Solve for x and the length of AC.

A B

CD

31. ABCD is a parallelogram. AD = 6x 2 and BC = x + 2 . Solve for x and the length of AD


Minimum Information to Prove Properties of quadrilaterals (with marked pictures)

Quadrilateral

Parallelogram: has been proven to be a quadrilateral and...

Rectangle: has been proven to be a parallelogram and...

Rhombus: has been proven to be a parallelogram and...

Square: has been proven to be a rectangle and a rhombus...

Trapezoid: has been proven to be a quadrilateral and.........

Isosceles Trapezoid: has been proven to be a trapezoid and........


Minimum Information to Prove Properties of quadrilaterals (with words)

Quadrilateral

1)

Parallelogram: has been proven to be a quadrilateral and...

1)

2)

3)

4)

5)

Rectangle: has been proven to be a parallelogram and...

1)

2)

Rhombus: has been proven to be a parallelogram and...

1)

2)

Square: has been proven to be a rectangle and a rhombus...

Trapezoid: has been proven to be a quadrilateral and.........

1)

2)

Isosceles Trapezoid: has been proven to be a trapezoid and........

1)

2)


Best Description using the minimum amount of info

Problems: For each of the following pictures decide whether or not it is a parallelogram, rhombus, rectangle or square. Also state the reason for your conclusion.

1. This one is done for you:

90°

A. It is a parallelogram because the opposite sides are equal. B. It is a rhombus because two adjacent sides are equal. C. It is a rectangle because it has a right angle. D. It is a square because it is a rectangle and a rhombus.

2.

90° 90°

90° 90°

3.


4.

5.

6.


7.

8.

9.

10.


11.

12.

13.

14.

15. sure about #14?


Best description in the coordinate plane

What is the best name for quadrilateral ABCD?1. A(–3, 8) B(0, 4) C(8, 10) D(5, 14)

2. A(–6, 4) B(10, 8) C(7, 3) D(–5, 0)


3. A(–3, 0) B(–2, –2) C(4, 1) D(3, 3)

4. A(–2, 5) B(–7, 2) C(–2, –1) D(3, 2)


5. A(0, 0) B(3, 1) C(1, 4) D(–2, 3)

6. A(–4, 5) B(2, –3) C(–1, 12) D(–4, 7)


7. A(1, 4) B(4, 9) C(–1, 12) D(–4, 7)

8. A(0, 0) B(12, 0) C(10, 6) D(2, 6)


9. A(1, 2) B(2, 5) C(5, 7) D(4, 4)

10. A(1, 1) B(5, 3) C(7, 7) D(3, 5)


What is the most specific name for each quadrilateral? You don’t need to list reasons.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14.


15. 16. 17.

18. 19. 20.

21. 22. 23.

24. 25. 26.

27. 28.


Documents

quadrilaterals - Lincoln-Sudbury Regional High School S2/Quads... · Quadrilaterals Tree of quadrilaterals Quadrilateral ... has all properties of a quadrilateral and ... K. May a