Geometry Unit #10 – Second Semester 2013 Quadrilaterals and Radicals Thursday – Jan 24

Parallelograms

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Friday – Jan 25

Proving Quads are Parallelograms, Rectangles and Squares

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Monday– Jan 28

Rhombus, Kite, Trapezoids, Isosceles Traps

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Tuesday– Jan 29

Trapezoids and Isosceles Trapezoids Medians and Midsegments

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Wednesday – Jan 30

Proofs, Simplifying and Multiplying Radicals

Thursday– Jan 31

Review

Friday- Feb 1

TEST – Quads and Radicals

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QUIZ HMWK – Packet Worksheet

Parallelograms -Jan 24 Classify each statement as true or false. ______1. Every quadrilateral is a parallelogram.

______7. There exists a parallelogram with all angles .

______2. Every parallelogram is a quadrilateral.

______8. WY is a diagonal of

______3. If quadrilateral RSTW is a parallelogram, then RS  TW.

______9. If AC is drawn in

WXYZ.

ABCD, then  ABC   CDA.

______4. If quadrilateral DEFG is a parallelogram, then DE // EF. ______5. There exists a parallelogram with all sides congruent. ______6. There exists a parallelogram ABCD such that m  A = 70 and m  C = 80. Exercises 10 - 17 refer to 10. RS // _____

RSTW. Complete each statement. 13. WRS  _____ 15. TX  _____

T

W X

14. RST  _____

11. TS  _____ 12. WX  _____

S

R

Exercises 16 - 19 refer to 16. CE = 12, CX = ______

CDEF.

F

19. m1 + m2 = 106, mFCD = ______

1

E 2

X

17. m3 = 88, m2 = ______ 3 C

18. mCFE = 41, mFED = ______ In each parallelogram, find the indicated measure. 20. Find AD. 4x + 3 A B

21. Find HI.

H

5x + 4

4

D

I

2x + 1 D

22. Find mR.

7x - 12 Q

C

K

J

3x + 36

23. Find QZ. XZ + YW = 52

R 4x + 8

X

Y 3a - 5

4a - 10

T

6x + 12

Q

W

S

Z

Use ABCD to answer the following. _____________24. Does AB + BC = AD + DC? A

_____________25. If mBAC = 37 and mBCA = 74, what is mADC?

X

_____________26. Is point X the midpoint of both AC and BD? _____________27. If AB = 4x  5, DC = 2x + 15, and BC = 4, then the perimeter of

ABCD is _______.

_____________28. If AC = 6x + 14 and XC = x + 15, then the length of AX is _______. _____________29. If mDAB = 5x  7 and mABC = 4x + 7, then mDCB = _______.

B

D

C

Given

KLMN. (Problems #30 - 33)

30. If KL =

x , MN = 2x  9, KL = ___________. 2

x2 , x = ___________. 2 3x 33. If mL = x  40, mN = , mL = _________. 4

31. If KL = 8, MN =

32. If mK = 31, mM = 2x2  1, x = __________. 34. Given:

ABCD with AB = 2x, CD = 3y + 4, BC = x + 7, and AD = 2y. Find the lengths of the sides of the parallelogram. A

35. DE = x + y BE = 10 AE = x  y CE = 8 Find x and y if ABCD is a parallelogram.

B E

D

C

Problems #34 and 35 ____________________________________________________________________________________________________________________

Proving Parallelograms, Rectangles, Squares - Jan 25 State whether the information given is sufficient to prove quad SMTP is a parallelogram. If so, give a reason. 1. SPT  SMT S

2. SPX  TMX, TPX  SMX

M X

3. SM  PT  SP  MT

P

T

4. SP  MT, SP // MT Find the value of x. Tell whether the figure must be a parallelogram. 3x − 2

5) 2x − 4

6.) x+4

2x + 6

7. WXYZ is a rectangle. The perimeter of rectangle WXYZ is 48. XY + YZ = 5x  1 XZ = 13  x Find WY.

(12x + 8) (2x)

(5x +2)

8. Rectangle ABCD has vertices A(3, 4) B(1, 6) C(5, 2) and D(1, 4). Find: a. the midpt of BD b. AB c. the slope of BC

9. SQRE is a square. The diagonals of SQRE intersect at A. Find: a. m  RSQ b. m  EAR c. EA = 5x  3, RA = 4x + 6. Find EQ. 10. WXYZ is a square. WX = 1  10x, YZ = 14 + 3x Find WY.

d. mRAE = 3(x − 10). Find x.

11. ABCD is a rectangle. F lies on BC. E lies on DC. mBAF = 29 and mDAE = 39. Find mFAE.

12. In rectangle ABCD, AB=15 and BC=6. Find the length of the diagonal. 13. In rectangle ABCD, diagonals AC and BD intersect at E. If AE = 2x − 6y, EC = 2x + 6, and BD = 16, find x & y. 14. In rectangle ABCD, mBAD = 2(mDAC) + 38 . Find mBAC. 15. The figure shows two similar rectangles. What is the length of PQ? A) 2 B) 3 C) 4 D) 5

Rhombus and Kite, Traps, Isos Traps - Jan. 28 1. Given: ABCD is a rhombus. m  3 = 56 Find: m  1 _______ m  4 _______ m  7 _______

m  10 _______

m  2 _______ m  5 _______ m  8 _______

m  11_______

m  3 _______ m  6 _______ m  9 _______

m  12_______

B 3 7 A

4 8 5 9

1 2

11 12

C

6 10 D

WXYZ is a rhombus. (Prob. #2, 3) 2. mX = 24(10  x), mZ = 6(x + 15), find mY.

3. WX = 3x + 2, XY = 5x − 10, find YZ.

4. Answer always, sometimes, or never. a. If a quadrilateral is a rhombus, then it is _______________ a square. b. If a quadrilateral is a square, then it is ________________ a rectangle. c. If a rectangle is a rhombus, then it is _________________ a square. d. If a quadrilateral is a rhombus, then it is __________________ a regular polygon. 5. Answer true or false. a. Every rectangle is a parallelogram. c. Every rhombus is a regular polygon.

b. The diagonals of a rhombus are perpendicular. d. If a rectangle is equilateral, then it is a square. R

6. Given: AROW is a kite m1 = 40 m10 = 30 AO = 10 RO = 13 Find: m2 _______

1 2

3 A 7

m6 _______

M 4 5 8 11

6 12 O

m11 _______

m3 _______

m7 _______

m12 _______

m4 _______

m  8 _______

MO = _______

m5 _______

m  9 _______

RM = _______

9 10

W

7. What is the perimeter of the kite below? x-4

I

9. If mKIE = 50 and mKEI = x + 5(x − 2), find x. x+3

17

K

KITE is a kite. (Probs. #8, 9) 8. If m11 = 4x and m10 = x + 10, find m8.

10

8 11

T

E

10. ABCD is a rhombus with diagonals AC and DB intersecting at R. If mBRC = 2x2 + 40, find x.

11. ABCD is a rhombus with diagonals AC and DB intersecting at R. If mADB = 2x − 1, mARB = 6x, mACB = y, find x and y.

12. In the isosceles trapezoid, m  A = 70. Find the measures of the other angles.

S

G

A

I

Problems #12 and 13 13. In the isosceles trapezoid, m  A = 5k. Find the measures of the other angles in terms of k.

14. Given: Isosceles trapezoid ABCD mBAC = 30, mDBC = 85 Find:

m1 _______

m6 _______

mDAB _______

m2 _______

m7 _______

mCBA _______

m3 _______

m8 _______

m4 _______

mADC _______

m5 _______

mBCD _______

B

A 8 3 D

7 5

6

1

2

4 C

____________________________________________________________________________________

Trapezoids, Isosceles Traps, Medians and Midsegments Jan. 29 Find the value(s) of the variable(s) in each isosceles trapezoid. 1. 2. 3x − 3 (6x+20)°

3.

Y°

7x

2x+5

(4x)°

x−1

x+5 Each trapezoid is isosceles. Find the measure of each angle. 4.

5. 3

2

77°

1 105°

Find the value of the variable in each isosceles trapezoid. 6.

7. 45°

60°

(3x+15)°

8.

T

U

TV = 2x − 1 US = x + 2

3x°

S

V

9. Given: Isosceles trapezoid JXVI mIXV = 83, mVJX = 28 Find: m1 _______

m6 _______

m10 _______

m2 _______

mIVX _______

m11 _______

m3 _______

m7 _______

m12 _______

_______

m8 _______

mVXJ _______

m5 _______

m9 _______

m4

J

RS is the median of trapezoid ABCD. 14. If AB = 10 and DC = 8, RS = __________

N

A

C

18. If RS = 7, then AB + DC = ___________ D

15. If BC = 12, CS = __________

V

B

M

13. MN = p, AC = ____________

11 12

3 4

I

In exercises 10 - 13, points M and N are the midpoints of AB and BC. 10. AC = 12, MN = ___________ 12. MN = 7, AC = ____________ 11. AC = k, MN = ____________

X 2 1 6 9 10 5 7 8

19. If m A = 80, mDRS = ___________

C

R

S

A

B

16. If DC = 2x + 8, RS = 4x + 18 and AB = 10x + 20, find RS.

17. If RS = 20 and DC = 14, then AB = ______________

Trap MNRS is isosceles with median XY. Find the following. 20. NX = 10, find MS. 22. mN = 40, find mS.

21. mR = 55, find mMYX.

Points R and S are the midpoints of AB and BC. 23. If RS = 5.6, AC = _________

B

R

24. If AC = 3x + 1 and RS = x + 3, then RS = __________

S

A

C



Quad Proofs , Simplifying Radicals - Jan 30 1. Given: ABCD is a rectangle ACBE is a parallelogram Prove: DB  EB Statements

D

C

A

B

E

Reasons_____________________________

1. ABCD is a rectangle

1. _________________________________________

2. DB  AC

2. _________________________________________

3. ACBE is a parallelogram

3. _________________________________________

4. AC  EB

4. _________________________________________

5. DB  EB

5. _________________________________________

2. Given: ABCD is a trapezoid with AB // CD AP bisects DAB Prove: APD is isosceles

P

D

C

B

A

Statements 1. ABCD is a trapezoid with AB // CD, AP bisects DAB

Reasons_____________________________ 1. _________________________________________

2. DPA  PAB

2. _________________________________________

3. DAP  PAB

3. _________________________________________

4. DPA  DAP

4. _________________________________________

5. AD  PD

5. _________________________________________

6. APD is isosceles

6. _________________________________________

F

C

3. Given: ABCD , FG bisects DB Prove: DB bisects FG Statements 1.

ABCD, FG bisects DB

D

E B

G

A

Reasons_____________________________

1. _________________________________________

2. CD // BA

2. _________________________________________

3. CDB  ABD, DFE  BGE

3. _________________________________________

4. BE  DE

4. _________________________________________

5. BEG  DEF

5. _________________________________________

6. FE  GE

6. _________________________________________

7. DB bisects FG

7. _________________________________________

CTGD, CO  DG, AG  CT 4. Given: Prove: COD  GAT

C

D Statements

O

A

T

G

Reasons_____________________________

1. CO  DG, AG  CT

1. _________________________________________

2. COD and TAG are right angles

2. _________________________________________

3. COD  TAG

3. _________________________________________

4.

CTGD

4. _________________________________________

5. DC  GT

5. _________________________________________

6. D  T

6. _________________________________________

7. COD  GAT

7. _________________________________________

R 5. Given: Isosceles Trapezoid RSPT Prove: TPQ is isosceles

S

T

P

Q Statements

Reasons_____________________________

1. Isosceles Trapezoid RSPT

1. _________________________________________

2. R  S

2. _________________________________________

3. RS // TP

3. _________________________________________

4. R  PTQ, S  TPQ

4. _________________________________________

5. PTQ  TPQ

5. _________________________________________

6. TQ  PQ

6. _________________________________________

7. TPQ is isosceles

7. _________________________________________

Simplify. 6.

27

7.

121

8.

800

9.

180

10.

100

11.

150

12.

117

13.

576

14.

320

15.

1100

16.

30

17.

44

18.

92

19.

32

20.

60

Multiply and simplify. 21.

3  15

22. 6 2 

3

25. 2 3  4 3

26. 4 6  2 3  5 6

29. 2 2  3 8

30.

33.

32 

2

6  2 12

23. 5 6  2 3

24.

27. 5 3  6 10

28. 3 5  4 5

31.

18  3 3

18 

25

32. 3 3  6 6

Review Quad Properties – Jan. 31 1. Given: Parallelogram ABCD with m  2 = 32, m  6 = 22, and m  11 = 61. Find: A m  1_________ m  7_________ m  3_________

m  8_________

m  4_________

m  9_________

m  5_________

m  10_________

11

7

8

10 1

6

9 3

4 5

2

D

B

C



2. Given: Rectangle RECT. m 2 = 49 Find: m  1_________ m  8_________ m  3_________

m  9_________

m  4_________

m  10_________

m  5_________

m  11_________

m  6_________

m  12_________

R

E 10

9

11 8

7 1

5

12

6 4

3

2

C

T

m  7_________ 

3. Given: Rhombus ABCD. m  4 = 33. Find: m  1_________

m  8_________

m  2_________

m  9_________

m  3_________

m  10_________

m  5_________

m  11_________

m  6_________

m  12_________

A 6

1

10 8

5

B 2

9 3 7 12 4 C

11

D

m  7_________ 

4. Given: Square SQUA. Find: m  7_________ m  1_________ m  2_________

m  8_________

m  3_________

m  9_________

m  4_________

m  10_________

m  5_________

m  12_________

m  6_________

m  12_________

S

Q 8

6

7

5

12 9

11 10

1

A

2

3

4

U



K

5. Given: Kite KITE. m12 = 52, m9 = 41. Find: m  1_________ m  7_________ m  2_________

m  8_________

m  3_________

m  10_________

m  4_________

m  11_________

m  5_________

1 2

I

4 3

5 6 8 7

12 11

T

9 10

E

m  6_________ 6. Put a checkmark in the “I Agree” column by the statements that you think are true. For any statement that is false, change any of the italicized words that you need to in order to make it true. I Agree 1.

Statement The hypotenuse is the shortest side of a right triangle.

2.

The diagonals of a kite are perpendicular.

3.

The consecutive sides of a parallelogram are congruent.

4.

The median of a trapezoid is found by adding the base lengths.

5.

A square is sometimes a rhombus.

6.

An isosceles trapezoid has diagonals that bisect each other.

7.

The consecutive angles in a parallelogram are congruent to each other.

8.

A rhombus has four congruent sides.

9.

A rectangle is always a square.

10.

A rhombus has congruent diagonals.

7. An open area at a local high school is in the shape of a quadrilateral. Two sidewalks crisscross this open area as diagonals of the quadrilateral. If the walkways cross at their midpoints and the walkways are equal in length, what is the shape of the open area? A) a parallelogram

B) a rhombus

C) a rectangle

D) a trapezoid