Download Assignments Quadrilaterals

Geometry Assignments: Quadrilaterals
Day
Topics
Homework
1
Quadrilaterals
HW Quadrilaterals – 1
2
Properties of parallelograms I
HW Quadrilaterals – 2
3
Properties of parallelograms II
HW Quadrilaterals – 3
4
Practice
HW Quadrilaterals – 4
5
Proving quadrilaterals are parallelograms
HW Quadrilaterals – 5
6
Rectangles
HW Quadrilaterals – 6
7
Rhombi & Squares
HW Quadrilaterals – 7
8
Practice
HW Quadrilaterals – 8
9
Trapezoids
HW Quadrilaterals – 9
10
Proofs practice
HW Quadrilaterals – 10
11
Review Day
HW Quadrilaterals – Review
12
**TEST**
HW
Grade
Quiz
Grade
Geometry HW Quadrilaterals – 1
1. The measures of the interior angles of a quadrilateral are in the ratio 2:3:4:6. Find the measures of all four
angles.
2. In quadrilateral ABCD, mA = x2, mB = 12x, mC = 8x – 5 and mD = 2x + 45. Find the measures of
all four angles of ABCD.
3. Quadrilateral PLGM has vertices P(0, 2), L(6, 6), G(10, 4) and M(4, 0).
a. Prove using coordinate geometry that both pairs of opposite sides of PLGM are parallel. What kind of
quadrilateral is PLGM?
b. Prove using coordinate geometry that both pairs of opposite sides of PLGM are congruent.
D
4. Parallelogram ABCD shown in the diagram has an area of 50. Find its
perimeter to the nearest tenth. (Area of a parallelogram is A = bh.)
C
x+2
A
x E
x+4
B
5. The definition of a parallelogram is: A parallelogram is a quadrilateral with both pairs of opposite sides
parallel. We wish to prove the following: Opposite sides of a parallelogram are congruent and opposite
angles of a parallelogram are congruent.
Given: Parallelogram ABCD with diagonal BD
a. Prove: AB  CD , BC  AD , A  C
b. Explain how you could prove B  D
6. In parallelogram ABCD, AB = 2x + 3, BC = 3x – 2 and CD = 4x – 10. Find the numerical value of the
perimeter of ABCD.
7. In parallelogram PARL, mP =
4
x  25 and mR = 5x  3 . Find the measures of all four angles of
3
PARL.
8. Quadrilateral FISH is rotated 90 around F. The image of I is O, the
image of S is W and the image of H is L. Is quadrilateral FISH congruent
to quadrilateral FOWL? Justify your answer.
W
L
O
F
I
S
H
Geometry HW Quadrilaterals – 2
Solve #1 – 4 algebraically.
1. In parallelogram ABCD, AB = x, BC = x + y, CD = y – x, and AD = 15. Solve for x and y.
2. In parallelogram GRAM, mG = x2 and mR = 12x + 20. Find the measures of all four angles of GRAM.
3. Two sides of a parallelogram measure 5 and 9.
a. Is it possible that the area of the parallelogram is 40?
b. Is it possible that the area of the parallelogram is 45?
c. Is it possible that the area of the parallelogram is 50?
4. In each diagram, assume the quadrilateral that looks like a parallelogram is one and that points that appear
collinear are collinear. Copy each diagram onto your paper and find the value of the variable(s).
a.
x
100
35
b.
c.
70
110
y
x
x
5. In parallelogram ABCD, diagonal AC is congruent to side AB and mBAC = 30. Find the value of
a. mB
b. mBCD
6. A parallelogram has a base that measures 6 less than twice its height. If the area of the parallelogram is 80,
find the length of its base.
7. In the diagram, ABCD is a parallelogram and side AB is extended to P
so that BP  CP . If mP = x and
mD = 3x + 10, find the value of x. (Hint: while not absolutely
necessary, it is helpful to give CBP a name such as y.)
C
D
(3x + 10)
x
A
B
P
8. Quadrilateral ABCD has vertices A(a, 0), B(b, h), C(b – a, h) and D(0, 0) where a, b and h are all positive
numbers.
a. Sketch ABCD on your homework paper.
b. Prove using coordinate geometry that ABCD is a parallelogram.
c. Prove using coordinate geometry that both pairs of opposite sides of ABCD are congruent.
9. Given: Parallelogram PQRS, QWVS , PV  QS , WR  QS .
Prove: QPV  SRW
P
Q
W
S
V
R
Geometry HW Quadrilaterals – 3
1. a. List two important properties of the sides of a parallelogram.
b. List two important properties of the angles of a parallelogram.
c. List one important property of the diagonals of a parallelogram.
2. In parallelogram QDRC, QD = x2 + 2, DR = x + 6 and RC = 4x + 14. Find the numerical length of the
perimeter of the parallelogram.
3. In parallelogram ABCD, mA = x, mB = 6y – 2x, and mC = 2y – 40. Find the values of x and y.
4. In parallelogram ABCD, diagonals AC and BD intersect at E. If BC is twice AB, AE is three more than AB,
BE is one less than AB and CE is 8 more than half of AB, find
a. the lengths of both diagonals of the parallelogram.
b. the perimeter of the parallelogram.
5. The diagonals of a parallelogram measure 10 and 16. What is the range of possible lengths for one side?
6. Two vertices of parallelogram PGRM are P(3, 2) and G(4, 6). The diagonals intersect at A(7, 5). Find the
coordinates of R and M.
7. Any parallelogram can be represented in coordinate geometry by the vertices A(a, 0), B(b, h), C(b – a, h)
and D(0, 0) where a, b and h are all positive numbers. Prove using coordinate geometry that the diagonals
of this (and therefore any) parallelogram bisect each other.
8. When quadrilateral ABCD is reflected over AB , the image of C is E.
a. What additional information would be needed to prove that ABCD 
ABEF?
b. With the additional information from part a, explain why the
quadrilaterals are congruent.
A
F
D
C
E
B
9. Given: Parallelogram ABCD, APB , DQC , AP  QC
P
A
B
Prove: AQ  PC
D
Q
C
Geometry HW Quadrilaterals – 4
1. In a certain parallelogram, two consecutive sides (that means not opposite sides) have measures x2 and
4x – 1 and the perimeter is 88. Find the lengths of the sides.
A
2. In parallelogram ABCD at right, E is a point on BD such that
AE  BD , mDAE = 50, and mBDC = 30. Find mC.
B
E
D
C
A
3. In the diagram at right, ABCD is a parallelogram.
Find the values of x, y and z.
(3x)
x
D
B
(5x)
z
y
C
4. In parallelogram PAIN, diagonals PI and AN intersect at S. If PS = x2 – 6, AS = 2x – 1, IS = 3y + 1 and
NS = x + y, find the numerical length of both diagonals.
5. We want to prove the following theorem: If both pairs of opposite sides of a quadrilateral are congruent,
the quadrilateral is a parallelogram.
Given: Quadrilateral ABCD with diagonal BD , AB  CD and AD  BC
Prove: ABCD is a parallelogram (using the definition of a parallelogram)
6. We want to prove the following theorem: If one pair of opposite sides of a quadrilateral is both parallel and
congruent, the quadrilateral is a parallelogram.
Given: Quadrilateral ABCD with diagonal BD , AB || CD and AB  CD
Prove: ABCD is a parallelogram
7. We want to prove the following theorem: If the diagonals of a quadrilateral bisect each other, the
quadrilateral is a parallelogram.
Given: Quadrilateral ABCD; AC and BD bisect each other at E.
Prove: ABCD is a parallelogram
Hint: Use either of the theorems from problem #5 or #6 above.
Geometry HW Quadrilaterals – 5
1. In the figures below, the givens are marked on the diagram (arcs indicate congruent angles, slashes indicate
congruent segments, and arrows indicate parallel segments). Tell whether the figure must be a
parallelogram and if so, give a specific reason.
a.
b.
c.
d.
e.
f.
g.
h.
2. In quadrilateral ABCD, mA = x, mB = x + 70, mC = 2x – 50, and mD = 2x + 10. Find the value of x
and determine whether ABCD is a parallelogram. Give a reason.
3. Quadrilateral ABCD has perimeter 80. If AB = x, BC = 2x – 5, CD = 2x – 15 and DA = x + 10, find the
value of x and determine whether ABCD is a parallelogram. Give a reason.
4. In quadrilateral ABCD, diagonals AC and BD intersect at E. AE = 4 and AC = 8, BE = x + 3 and
BD = 2x + 6.
a. Is it possible to solve for x? Demonstrate.
b. Is it possible to determine if ABCD is a parallelogram? If so, how? If not, why not?
5. Prove that if a quadrilateral has one pair of opposite sides parallel and one pair of opposite angles
congruent, then the quadrilateral is a parallelogram.
Given: Quadrilateral ABCD with diagonal BD , AB || CD and A  C
Prove: ABCD is a parallelogram
6. One way to prove a quadrilateral is a parallelogram using coordinate geometry is "Show both pairs of
opposite sides have the same slope and are thus parallel." List three other ways to prove a quadrilateral is a
parallelogram using coordinate geometry.
7. Definition: A rectangle is a quadrilateral with four right angles.
Given: Rectangle ABCD
Prove: a. ABCD is a parallelogram (This can be done in three or four steps.)
b. AC  BD
Geometry HW Quadrilaterals – 6
1. In rectangle ABCD, the diagonals AC and BD intersect at E. If AB = 8 and BC = 6, find the values of
a. AD
b. CD
c. AC
d. BE
2. In parallelogram ABCD, the diagonals AC and BD intersect at E, AE = 7x – 1, CE = 5x + 5, and
DB = 10(x + 1).
a. Find the value of x
b. Find AC
c. Find DB
d. Is ABCD a rectangle? Give a reason.
3. In parallelogram ABCD, mA = 2x + 20 and mB = 4x – 50. Find the value of x and determine whether
ABCD is a rectangle. Give a reason.
4. In rectangle ABCD, the diagonals AC and BD intersect at E. If AE = 3x + y, BE = 4x – 2y,
CE = 20, find the values of x and y.
5. Which of the following is not sufficient to prove a quadrilateral is a parallelogram?
(1) Both pairs of opposite sides 
(2) One pair of opposite sides || and 
(3) Its diagonals are .
(4) Its diagonals bisect each other.
6. Which of the following is sufficient to prove a quadrilateral is a rectangle?
(1) It has one right angle.
(2) Its diagonals bisect each other.
(3) Its diagonals are congruent.
(4) It is equiangular.
7. Which of the following is sufficient to prove a parallelogram is a rectangle?
(1) Its diagonals bisect each other.
(2) Its diagonals are congruent.
(3) Its diagonals are perpendicular.
(4) Its diagonals bisect its vertices.
8. Quadrilateral RECT has coordinates R(–3, 2), E(6, 8), C(10, 2) and T(1, –4). Prove using coordinate
geometry that RECT is a rectangle.
9. The vertices of quadrilateral WORM are W(10, 6), O(0, 0), R(a, 5) and M(a + 10, b).
a. Find the value of b that will make WORM a parallelogram.
b. Using the value of b from above, find the value of a that will make WORM a rectangle.
M
A
10. Given: Rectangle ABCD, m is the midpoint of side AB .
Prove: a. DM  CM
b. MDC  MCD
D
B
C
Geometry HW Quadrilaterals – 7
1. Draw rhombus ABCD with diagonals AC and BD intersecting at E. For each of the following, tell if the
statement is true or false and if the statement is true, give a reason.
a. AB  AD
b. AE  EC
c. AC  BD
d. AB || CD
e. DAB  DCB
f. DAC  BAC g. AEB  AED
h. EAD  ECB
i. EAB  ECB
j. DAB and ABC are supplementary
2. Find the values of the variables
in the diagrams at right.
a.
z
b.
y
65
x
y
25
z
x
3. In square ABCD, diagonal AC is drawn.
a. Find mBAC and mBCA and justify your answer.
b. If AB = 1, find the value of AC.
4. In a rhombus, the length of a side is 6 and the length of the shorter diagonal is 8. Find the length of the
longer diagonal.
5. Quadrilateral RHMB has vertices R(0, 0), H(a, b), M(a + b, a + b) and B(b, a) where a and b are both
positive numbers. Prove using coordinate geometry that RHMB is a rhombus.
6. Quadrilateral LYNX has vertices L(–20, 12), Y(28, 48), N(64, 0) and X(16, –36).
a. Prove that LYNX is a parallelogram.
b. Find the lengths of the diagonals of LYNX. Is LYNX a rectangle? Justify your answer.
c. Find the slopes of the diagonals of LYNX. Is LYNX a rhombus? Justify your answer.
d. What is the most descriptive term for LYNX?
Geometry HW Quadrilaterals – 8
1. The following may be review. Or may not. A set is a well-defined collection of things called elements. A
subset is part (or possibly all) of another set. If B is a subset of A, then all the elements in B are also in A
but there may be elements in A that are not in B. On a Venn diagram, if one circle is completely inside
another circle, the inside one is a subset of the outer one.
Ex: The consonants are a subset of all the letters of the alphabet.
The odd numbers are a subset of the integers.
The primes are not a subset of the odds (because of 2).
Dogs are a subset of mammals.
Poodles are a subset of dogs.
“Furry four-legged animals with tails” is not a subset of dogs.
True or false:
a. Quadrilaterals are a subset of parallelograms.
b. Rectangles are a subset of parallelograms.
c. Rhombi are a subset of rectangles.
d. Squares are a subset of rectangles.
e. Squares are a subset of rhombi.
f. Rectangles are a subset of rhombi.
(This assignment is continued on the next page.)
2. In quadrilateral ABCD, if AB  CD and AD  BC , then diagonals AC and BD must
(1) be perpendicular.
(3) be parallel.
(2) be congruent.
(4) bisect each other.
3. Which of the following is not sufficient to prove a quadrilateral is a parallelogram?
(1) Both pairs of opposite sides are congruent (2) Both pairs of opposite angles are congruent
(3) The diagonals are congruent.
(4) The diagonals bisect each other.
4. Which is not sufficient to prove a quadrilateral is a rectangle?
(1) It has four right angles.
(2) Its diagonals are congruent.
(3) It is a parallelogram with a right angle.
(4) All pairs of consecutive sides are perpendicular.
5. Which is not sufficient to prove a quadrilateral is a rhombus?
(1) It has all sides congruent.
(2) Its diagonals are perpendicular and bisect each other.
(3) It is a parallelogram with two consecutive sides congruent.
(4) It is a parallelogram with congruent diagonals.
6. In the diagram at right, CDE and AD bisects BAE,
mDAB = 40, mABC = 60, and mBCD = 120,
a. Explain how we know AB CDE
b. Find mE.
D
C
E
120
B
60
40
A
7. In rhombus ABCD, diagonals AC and BD intersect at E. If mAEB = 3x + y and
1
mCED = 4x – 3 y, find x and y.
B
8. Find the values of x, y and z in the diagram at right.
C
y
z
D
D
P
A
9. Given: Parallelogram ABCD, APB , DQC , DAQ  BCP.
Prove: a. DAQ  BCP
b. Quadrilateral APCQ is a parallelogram.
x
40
A
E
Q
10. Quadrilateral WHAT has vertices W(1, 0), H(0, 3), A(6, 5) and T(7, 2).
a. Determine using coordinate geometry if the diagonals of WHAT bisect each other.
b. Determine using coordinate geometry if the diagonals of WHAT are congruent.
c. Determine using coordinate geometry if the diagonals of WHAT are perpendicular.
d. What kind of quadrilateral is WHAT and why?
B
C
Geometry HW Quadrilaterals – 9
1. Lean this vocabulary:
a. A trapezoid is a quadrilateral with at least one pair of opposite sides
parallel.
b. The parallel sides are called bases. (If both pairs of opposite sides are
parallel, either pair can be the bases.)
c. The other pair of sides are called legs.
Bases
A
B
D
C
Legs
2. Which of the following are trapezoids? (Select all that are trapezoids.)
(1)
(2)
(3)
(4)
3. Determine if each of the following is a trapezoid and explain why or why not.
a.
b.
c.
114
65
d.
65
4. In trapezoid ABCD, AB || CD .
a. Which pairs of angles, if any, must be congruent?
b. Which pairs of angles, if any, must be supplementary?
5. An isosceles trapezoid is a trapezoid with a line of symmetry through its bases.
In the diagram, ABCD is a trapezoid with line of symmetry ℓ intersecting the
bases at P and Q.
a. What must points P and Q be?
b. Explain why A  B and C  D (base angles are congruent),
AD  BC (legs are congruent), and AC  BD (diagonals (not shown) are
congruent). (Think about what it means for a figure to have line symmetry.)
c. Explain why A and C are supplementary (B and D are supplementary
by the same reasoning) (opposite angles are supplementary).
ℓ
A
D
P
B
Q
6. Based on the definition in problem #5, is a rectangle an isosceles trapezoid? Justify your answer.
7. In the diagram at right, AB || CD .
B
A
30
a. Find the value of x.
70
b. Find the value of y.
z
c. Orville says that z must be 70 because alternate interior angles are
x
y
C
D
congruent. Determine if he is right or wrong and justify your answer.
d. Hortense says that z must be 50 because opposite angles in a trapezoid are supplementary. Determine if
she is right or wrong and justify your answer.
e. Can we find mA? Explain.
8. The following figures are both isosceles trapezoids (but not rectangles). Find the measures of the angles
indicated by variables.
a.
80
25
b.
z
z
x
y
x
80
y
Go to next page for
more fun trapezoid
problems.
C
9. By choosing the right values for a, b, c and d, the vertices O(0, 0), A(2a, 2d), B(2b, 2d) and C(2c, 0) can
represent any trapezoid.
a. Sketch trapezoid OABC. Assume 0 < a < b < c and d > 0.
b. Prove using coordinate geometry that the segment connecting the midpoints of the legs of OABC is
parallel to the bases.
c. Prove using coordinate geometry that the length of the segment connecting the midpoints of the legs of
OABC is the average of the lengths of the bases.
10. In trapezoid MORE, MO || RE , MO = 40 and ER = 24.
a. Find the length of the segment that joins the midpoints of the legs of MORE.
b Leg ME is extended past E to T such that ME  ET and
leg OR is extended past R to S such that OR  RS . Find the
length of ST in trapezoid MOST.
M
E
24
40
R
O
Geometry Homework: Quadrilaterals – 10
BD are drawn
a. Prove: ABE  CDE
b. If AB = 6, AE = 4, BE = 3 and CD = 15, find the
length of diagonal AC
E
C
D
2. Given: Quadrilateral ABCD, BFED , BAE  DCF,
AE  BD , CF  BD , DE  BF .
Prove: a. ABE  CDF
b. ABCD is a parallelogram.
3. Given: ABCD is a rectangle; APQB , DRQ , CRP ,
B
A
1. Given: Trapezoid ABCD with AB || CD , diagonals AC and
A
B
F
E
D
C
D
C
AP  BQ .
R
Prove: a. DQ  CP
b. PR  QR
A
P
Q
B
4. Triangle ABC has vertices A(–3, 1), B(–1, 4) and C(8, y)
a. Find the value of y that will make ABC a right angle. Justify why this value makes ABC a right
angle.
b. Find the coordinates of D so that quadrilateral ABCD is a rectangle. Using your coordinates, prove that
ABCD is a rectangle.
Geometry Review: Quadrilaterals
1. What is the sum of the angles in a quadrilateral?
2a.
b.
c.
d.
List the five main properties of a parallelogram.
List the two main properties of a rectangle beyond those listed in part a.
List the three main properties of a rhombus beyond those listed in part a.
What are the properties of a square?
3. Which of the following are not all parallelograms?
(1) trapezoids
(2) squares
(3) rhombuses
(4) rectangles
4. In parallelogram PQRS, diagonals PR and QS intersect at M. Which must be true?
(1) PR  QS
(2) PM  RM
(3) PQ  QR
5. In rectangle ABCD, AC is a diagonal. Which must be true?
(1) AB  AC
(2) AB  AD
(3) DAC  BAC
(4) SMP  QMP
(4) DAC  BCA
6. Which quadrilateral has diagonals that are always congruent but not always perpendicular?
(1) rectangle
(2) rhombus
(3) parallelogram
(4) square
7. Which is a quadrilateral whose diagonals are always perpendicular?
(1) rectangle
(2) rhombus
(3) parallelogram
(4) trapezoid
8. Which is a quadrilateral that is equiangular but not always equilateral?
(1) rectangle
(2) rhombus
(3) square
(4) parallelogram
9. Which of the following is false?
(1) A square is a rhombus.
(3) A rectangle is a parallelogram.
10. The diagonals of a rectangle are always
(1) congruent.
(3) parallel.
(2) A rhombus is a rectangle.
(4) A parallelogram is a quadrilateral.
(2) perpendicular.
(4) bisectors of the angles of the rectangle.
11. Find the perimeter of parallelogram ABCD if AB = x + 7, BC = 2x – 1, and CD = 3x  4.
12. In parallelogram ABCD, diagonals AC and BD intersect at E. If AE = x, BE = xy, CE = 4y, and
DE = 25, find the numerical lengths of both diagonals.
E
D
30
y
z
13. In isosceles trapezoid DEFG, DE || GF and DG  EF . Using the
angles given in the diagram, find the values of w, x, y, and z.
x
w
75
F
G
14. The perimeter of a rhombus is 36 and one diagonal measures 180 .
Find the length of the other diagonal.
L
15. In the diagram at right, FLIP is a parallelogram and R is a point on
FP . Find the values of x and y.
(This assignment is continued on the next page.)
x I
80
F
y
R
110
P
16. In the diagram at right, FLOP is a rectangle. Find the values of x and y.
L
O
x
4y  10
y
F
P
L
A
17. In the diagram at right, FLAP is a rhombus. Find the values of x, y, z and
mFLA.
z
y
18. In rhombus ABCD with diagonal AC , the ratio of mBAC to mD is 2:5.
Find the numerical value of mB.
19. In parallelogram ABCD, side BA is extended past A to E such that
AE  AD . If mE = x + 24 and mC = 5x – 12,
find the value of x.
20
x
P
F
A
E
B
(x + 24)
(5x – 12)
C
D
20. Quadrilaterals KITE and SOAR are graphed at right. Explain using the
properties of ridged motions why KITE  SOAR.
y
The following are true (meaning always true) or
false (meaning not always true). For the false ones, explain why they are
false or give a counterexample.
21. If one angle of a parallelogram is a right angle, then all the other angles
E
are also right angles.
22. If a quadrilateral is equiangular, then it is a square.
23. If parallelogram has line symmetry, then it is a rectangle.
24. The diagonals of a parallelogram divide it into four congruent triangles.
T
25. If the diagonals of a quadrilateral are congruent, then the quadrilateral is
a rectangle.
26. A quadrilateral with all four sides congruent must be a square.
27. If the diagonals of a quadrilateral are perpendicular then the quadrilateral is a rhombus.
R
S
O
A
K
I
x
Answers
1. 360
2a. Both prs opp sides ||; both prs opp sides ; both prs opp s ; all prs consec s supp; diags bis each other
b. All rt s (equiangular); diags 
c. All sides congruent (equilateral); diags ; diags bisect the vertices (angles)
3. (1)
4. (2)
5. (4)
6. (1)
7. (2)
8. (1)
9. (2)
10. (1)
11. 45
12. 20, 50
13. (30, 45, 105, 75)
14. 12
15. (70, 30) 16. (90, 20)
17. (20, 70, 90), 140
18. 100
19. 20
20. SOAR is the image of KITE after a translation. A translation is a rigid motion, meaning it preserves
distances (and angles) so the two quadrilaterals are congruent.
21. T
22. F; must be a rectangle
23. F; could be a rhombus
24. F; both pairs of “vertical” s are  but “adjacent” s won’t always be
25. F; that’s only if the quadrilateral is a parallelogram
26. F; it must be a rhombus
27. F; only true if the quadrilateral is a parallelogram.