geometry - niskayuna central school...

Geometry Chapter 5 Midsegment of a Triangle

Median of a Trapezoid

Quadrilaterals

Proofs

1

Name: _________________________________________________________________ Geometry Assignments – Chapter 5

Parallel Line Proofs & Chapter 5- Quadrilaterals

Date Due

Section

Topics

Assignment Written Exercises

2-2,

5-1 &

5-4

Parallelograms & Their Properties

Rectangles and Their Properties

Pg 169-170 #2, 5-12, 19-21, 24 & 25

Algebra in Geometry Worksheet - #3,

5, 7, 8

5-4 &

5-5

Squares & Their Properties Rhombuses & Their

Properties Trapezoids & Their Properties Isosceles Trapezoid & Their

Properties

Worksheet – “More Quads”

5-3 &

5-5

Midsegement of triangle

Median of trapezoid

Pg180 #1-9 odd, 16, 17 Pg 192 [written ex at bottom of page]

#1-9, 13-17 odd

5-2, 5-4

& 5-5

Using the Quadrilaterals to Prove Parts or Triangles

Worksheeet

5-2, 5-4

& 5-5

Proving the various Quadrilaterals

Worksheet

5-2, 5-4

& 5-5

More proofs Review

Worksheet “ Ch 5 Review of Quads”

2

Chap. 5

Review

Suggested Chapter 5 Review

Questions from your Textbook

Use the suggested Practice as a Guide, Ask for help if you have questions

Pg. 45 #10 (Classroom Ex.)

Pg. 46 #4, 7, & 8 (Written Ex.)

Pg. 182 #1-5, 7& 8 (Self Test 1)

Pg. 195 #1-7 all, (Self Test 2)

Pg. 197 #1-8, 13-22(Chap. Review)

Pg. 199 #9-16, 18 & 19 (Chap. Test)

All Answers for above questions are

available, simply ask for them!

Learn the Math by doing the math!

3

Chapter 5: Quadrilaterals Quadrilateral Properties

Parallelogram:

Property Yes No

Opposite Sides Parallel

Opposites Sides Congruent

Opposite Angles Congruent

Consecutive Angles Supplementary

All Sides Congruent

All Angles Congruent (all right angles)

Diagonals Congruent

Diagonals are Perpendicular

Diagonals Bisect Each Other

4

Rectangle:

Property Yes No





All Sides Congruent


Diagonals Congruent



5

Rhombus:

Property Yes No





All Sides Congruent


Diagonals Congruent



6

Square:

Property Yes No





All Sides Congruent


Diagonals Congruent



7

Trapezoid:

Property Yes No

Only one Pair Opposite Sides Parallel

Non-Parallel Sides Congruent

Base Angles Congruent



Diagonals Congruent



Diagonals Bisect the Angles

8

Isosceles Trapezoid:

Property Yes No

Only one Pair Opposite Sides Parallel

Non-Parallel Sides Congruent

Base Angles Congruent



Diagonals Congruent



Diagonals Bisect the Angles

9

Algebra in Geometry quads WS Name ________________________

Geometry Date __________ Block _______

Review Properties First!

List all the quadrilaterals that have each of the following properties.

A. Quadrilaterals with diagonals that bisect each other.

B. Quadrilaterals with only one pair of parallel sides.

C. Quadrilaterals with all pairs of consecutive angles congruent.

D. Quadrilaterals with congruent diagonals.

E. Quadrilaterals whose diagonals bisect the angles.

10

F. Quadrilaterals with perpendicular diagonals.

___________________________________________________________________

1. Two angles formed by one diagonal of a rectangle and the sides of the rectangle are 2 6 50x x and 10 8x . Find the measures of the two angles.

2. Two consecutive angles in a parallelogram are represented by 2 6 50x x and

2 6 50x x .

Find all possible measures for the angles.

3. Opposite sides in a parallelogram are represented by 212 10x x and

22 13 3x x . If x is

greater than one, find the lengths of the sides of the parallelogram.

4. Two consecutive angles of a parallelogram are represented by 22 50x x and

2 6 100x x . If x is an integer, find the measures of the angles.

5. Opposite angles in a parallelogram are represented by 225 5 4x x and 15x . Find the

measures of all angles of the parallelogram.

11

6. Two angles formed by one diagonal of a rectangle and the sides of the rectangle are 2 2 50x x and 5 50x . If the smaller angle is greater than 30, find the measures of the two

angles.

7. Two consecutive angles of a parallelogram are 2 2 150x x and 3 6x . Find the measures

of the angles.

8. Opposite sides in a parallelogram are represented by 2 45x and 2 35x . Find the two

possible lengths of the sides.

9. Two consecutive angles of a parallelogram are represented by 2 20 100x x and

2 9 59x x . If x is a positive integer, find the measures of the angles.

10. Opposite angles in a parallelogram are represented by 27 17 21x x and

23 12x x . To

the nearest degree, find the measures of all of the angles of the parallelogram.

11. Two angles formed by one diagonal of a rectangle and the sides of the rectangle are 210 18 80x x and

211 7x . Find the measures of the two angles.

12

More Quads WS Name __________________________

Geometry Date _________ Block _____

Answer the following questions. Show all algebra work when necessary (separate paper).

For what values of x and y is quadrilateral RSTW a parallelogram?

1. RS = 2x + 7, ST = 3y – 5,

TW = 25, WR = 16

2. WZ = 4x – 3, ZS = 13,

RZ = 17, ZT = 7y + 3

3. m<WRS = 24x, m<RST = 14y – 4,

m<STW = 15x + 27, m<TWR = 11y + 20

4. m<WRS = 75, m<RST = 7x,

m<STW = 11y + 9

________________________________________________________________________

Refer to parallelogram JKLM.

5. Name four pairs of congruent segments.

6. If m<MJK = 65, state the measures of <JKL,

<KLM, and <LMJ.

R

W T

Z

S

J

M L

N

K

13

7. If NJ = 7, JL = ______.

8. IF MK = 10, then NK = _____.

9. If m<MJL = 37 and m<LJK = 27,

then m<JKL = _____.

10. IF m<JMK = 71 and m<KML = 42,

then m<JKL = _____ and m<MKL = _____

Refer to parallelogram JKLM above. Find the value of x or y.

11. MJ = 2y + 5; LK = 14 – y 12. m<MJK = 4x + 4; m<KLM = 74 – x

13. m<MLK = 2x + 9; m<JKL = 5x + 3 14. JL = 4y + 6; NL = 3y – 1

15. MN = y + 4; MK = 5y – 10 16. m<MJL = 4x + 7; m<LJK = 5x – 8;

m<MLK = 7x + 13

Refer to rectangle RSTW.

17. What is m<RWT?

18. Find m<1 + m<2.

19. If m<1 = 3x + 12 and m<2 = 2x – 7,

find x and m<1.

R S

T W

Z

1

2

14

20. If RT = 5y + 2 and WS = 11y – 10,

find y and WS.

21. If RT = 7a – 2 and WZ = 4a – 3,

find a and RZ.

22. If m<1 = 61, what is m<RZW?

________________________________________________________________________

Refer to rhombus ABCD.

23. What is m<AEB?

24. If m<1 = 50, find m<2 and m<3.

25. If m<DAB = 7x + 14 and

m<2 = 5x – 5, find x and m<1.

26. If AD = 3w + 7 and AB = 2(w + 8),

find w and AD + DC.

27. If AC = 2y + 8 and EC = 2y – 1,

find y and AE.

28. If AB = 3k + 1 and the perimeter

of rhombus ABCD is 13k – 1, find AB.

A

B

C

D E

1

3

2

15

________________________________________________________________________

29. In a parallelogram TUVW, TU = 3z – 14 and WV = 2z – 6.

a. Find the value of z.

b. If UV = z + 2, what kind of parallelogram is TUVW? Why?

30. In a parallelogram TUVW with diagonals that intersect at X,

TX = 2y + 11, VX = y + 9, and WU = y + 18.

a. Find TV.

b. Can TUVW be a rectangle? Why?

31. In a parallelogram ABCD, AB = 2x + 3, BC = 4x – 5, and CD = 5x – 9. Show that ABCD is a

rhombus.

16

Sect. 5.3 – Midsegments of Triangles & 5.5 – Trapezoids

Recall: The median of a triangle is a line segment from the vertex of a

triangle to the midpoint of the opposite side.

Def – the midsegment of a triangle is a segment that joins

the ________________ of two sides of a triangle.

Theorem – A line that contains the midpoint of one side of a

triangle is parallel to another side passes through the

_____________ of the third side.

17

Theorem – The segment that joins the midpoints of two sides of a triangle

a.) is _________________ to the third side.

b.) is half as long as the _____________ side.

Def – The median of a trapezoid is the segment that joins the

______________ of the legs.

Theorem – The median of a trapezoid

a.) is _____________ to the bases.

b.) has a length equal to the _____________ of the base lengths.

18

Algebra Examples

Midsegments of Triangles and Medians of Trapezoids

Please use a separate piece of paper for all work!

1. Given triangle ABC with D & E midpoints of sides AB and AC respectively, find the value of x

if DE = 2x+1 and BC = 5x-5.

2. In trapezoid FGHI, FG IH and J is the midpoint of FI and K is the midpoint of GH . If JK

= 8, FG = x2+x-2 and IH = x2+3x-12, find the value of x.

3. In triangle MLN, J and K are midpoints of sides ML and MN respectively. If JK = 3x-1.5 and

LN = 15, find x.

4. In a trapezoid TSRQ, TS QR , U is the midpoint of TQ , and V is the midpoint of SR . If TS =

8x+34, UV = 86, QR = 14x+92, find the value of x.

5. In trapezoid ABCD, AD BC , M is the midpoint of AB and N is the midpoint of DC . If AD =

x2+1, MN = 4x+1, and BC = x2+2x+1, find AD, MN, and BC.

Ans: 1. 7 2. -3 3. 3 4. 23/11 5. AD = 10, MN = 13, BC = 16

19

Geometry Practice w/ prop of Quads.

Give the best name for the special quadrilateral described in each of the following

sentences for Quads ABCD.

1.

2. AB DC and AD BC

3. AC BD and and are perpendicular bisectors of each other.

4. , , and

_____________________________________________________________

True or False

5. The diagonals of a parallelogram always bisect opposite angles.

6. The diagonals of a parallelogram always bisect each other.

7. If only one pair of sides of a quadrilateral are parallel and the other pair of sides

are congruent, the quadrilateral is an isosceles trapezoid.

8. A square is a rectangle.

9. A square is a rhombus.

10. A square is a parallelogram.

11. A diagonal of a rhombus forms a pair of congruent triangles.

AC BD

AB DC AB BC AD AB

20

12. The parallel sides of an isosceles trapezoid are congruent

____________________________________________________________

13. In rectangle ABCD, AB = 25, BC = 16, CD = 3x + y, and AD = 5x - 2y. F

ind the values of x and y.

14. In rhombus RHOM, the diagonals and intersect at B. If m<BRH =

and m<BHR = , find x and m<BRM.

15. The diagonals of quadrilateral PGRA intersect at M. If PM = 9x + 17, GM = 3y +

29, RM = 12x + 5, and AM = 5y + 13, for what values of x and y would PGRA be a

parallelogram? Given these values, would PGRA be a special parallelogram?

Explain.

RO HM

2x + 3x + 9 2

2x - 9

21

Parallelogram

59. Parallelogram opp. sides parallel

60. Parallelogram opp. sides

61. Parallelogram opp. angles

62. Parallelogram diagonals bisect each other

63. Parallelogram consecutive angles supplementary

Rectangle

64. Rectangle opp. sides parallel

65. Rectangle opp. sides

66. Rectangle opp. angles

67. Rectangle diagonals bisect each other

68. Rectangle consecutive angles supplementary

69. Rectangle 4 right angles which are

70. Rectangle diagonals are

Rhombus

71. Rhombus opp. sides parallel

72. Rhombus opp. sides

73. Rhombus opp. angles

74. Rhombus diagonals bisect each other

75. Rhombus consecutive angles supplementary

76. Rhombus 4 sides

77. Rhombus diagonals are perpendicular

78. Rhombus diagonals bisect the angles

22

Square

79. Square opp. sides parallel

80. Square opp. sides

81. Square opp. angles

82. Square diagonals bisect each other

83. Square consecutive angles supplementary

84. Square 4 sides

85. Square 4 right angles which are

86. Square diagonals are

87. Square diagonals are perpendicular

88. Square diagonals bisect the angles

Trapezoid

89. Trapezoid one pair of opposite sides parallel

Isosceles Trapezoid

90. Isosceles Trapezoid one pair of opposite sides parallel

91. Isosceles Trapezoid non parallel sides

92. Isosceles Trapezoid base angles

93. Isosceles Trapezoid diagonals

Proving a quadrilateral is a parallelogram

94. both pairs of opposite sides parallelogram

95. both pairs of opposite sides parallelogram

96. 1 pair of opposite sides both and parallelogram

97. diagonals bisect each other parallelogram

98. both pair of opposite angles parallelogram

23

Proving a quadrilateral is a rectangle

99. parallelogram with 1 right angle rectangle

100. parallelogram with diagonals rectangle

101. quad w/ 4 right angles rectangle

Proving a quadrilateral is a rhombus

102. parallelogram with 2 adj. sides rhombus

103. parallelogram with diagonals rhombus

104. parallelogram w/ 1 diagonal that bisects opp. rhombus

105. a quadrilateral w/ 4 sides rhombus

Proving a quadrilateral is a square

106. rectangle w/ 2 adj. sides square

107. rhombus w/ 1 right angle square

Proving a quadrilateral is a trapezoid

108. quad. w/ 1 pr. opp. sides trapezoid

Proving a quadrilateral is an isosceles trapezoid

109. quad. w/ 1 pr. opp. sides and legs isosceles trapezoid

110. trapezoid with non parallel sides isosceles trapezoid

111. trapezoid with base angles isosceles trapezoid

112. trapezoid with diagonals isosceles trapezoid

24

1. Given: is a parallelogram;PQRS

PS QT 2. Given:

is a parallelogram;PQRS

QR QT

Prove: is isoscelesQRT Prove: S T

3. Given: is a parallelogram;JKLM

JO OL 4. Given:

is a parallelogram;JKLM

JP QL

Prove: OP OQ Prove: and bisect each otherJL QP

Q

R S T

P Q

R S T

P

J P K

M Q L

O

1

2

J P K

M Q L

O

1

2

25

________________________________________________________________________

5. Given: is a parallelogram;AECF

FD BE 6. Given:

is a parallelogram;ABCD

FD BE

Prove: AD BC Prove: AF EC

_______________________________________________________________________

7. Given: is a rhombus;

AC

ACEF

BC

Prove: 1 2

A B

C D

E

F

A B

C D

E

F

C A

E

B

F

2

1

26

8. Given: is a rectangle;

M is midpoint of

XYRS

YR 9. Given:

is a rectangle;

XM

XYRS

SM

Prove: XM SM Prove: M is midpoint of YR

10. Given: Parallelogram ABCD;

DE FB

Prove: EGC FGA

X Y

M

R S

X Y

M

R S

27

11. Given: Parallelogram FLSH;

LG FS ; HA FS

Prove: LGS HAF

12. Given: ABCD is a parallelogram;

AJ CE

Prove: GH FG

28

13.

14. Given: Isosceles trapezoid with BC AD, GP AB ,

EQ CD , P and Q are midpoints of AB and CD respectively.

Prove: APG EQD

29

15. Given: Trapezoid ABCD, BC AD , BE and CF are altitudes drawn to AD , AE DF

Prove: Trapezoid ABCD is isosceles.

16. Given: Isosceles trapezoid RSTW

Prove: RPW is isosceles

30

Quad Proofs Day II – Proving Quads

1. Given: ;AB DC AD BC 2. Given: ;P R Q S

Prove: 1 2 Prove: PQ RS

3. Given: ; ;

1 3

AK BJ BJ BL

4. Given:

is a parallelogram;JKLM

PX QX

Prove: is a parallelogramABJK Prove: is a parallelogramJPLQ

A

C

B

D 1

3

4

2

5 S

P

R

Q

B

J K L

A

2 1 3

J K

P X

Q

M L

31

5. Given: ;RX RY

RYS TZS

6. Given:

; ;RX RY RT XZ

RT XZ

Prove: is a parallelogramRTZX Prove: RYS TZS

7. Given: is a parallelogram;

AC ; 1 2

ACEF

BC 8. Given:

is a rhombus;

ABX

ABCD

BAX

Prove: is a rhombusACEF Prove: is a squareABCD

R S T

Y

X Z

R S T

Y

X Z

C A

E

B

F

2

1

A B

C D

X

32

9. Given: 1 2; 2 3;

3 4

10. Given:

1 2; 2 3;

PQ RS

Prove: is a rhombusPQRS Prove: is a rhombusPQRS

________________________________________________________________________

11. Given: parallelogram ABFE;

Parallelogram EFCD;

AD CD

Prove: ABCD is a parallelogram

12.

13.

S R

P Q

1

2 3

4 S R

P Q

1

2 3

4

33

14.

.

Prove: ABEF is a Rectangle

15. Given: Trapezoid ABCD, BC AD , EB EC

Prove: ABCD is an isosceles trapezoid

16. Given: ; ;AF CF BF DF

AB BE

Prove: CD BE

B A

C

E

F

D

34

17.

35

Chapter Five Review – Quadrilaterals

You must show all work and diagrams on separate paper.

Part I: Multiple Choice. Questions # 1-7, select the best possible answer.

1. A parallelogram must be a rhombus if the

a. diagonals are congruent c. diagonals are perpendicular

b. opposite angles are congruent d. opposite sides are congruent

2. Which one of the following statements is always true?

a. A quadrilateral is a trapezoid c. A rectangle is a parallelogram

b. A trapezoid is a parallelogram d. A rhombus is a square.

3. In parallelogram ABCD, diagonals AC and DB intersect at E. Which one of the following

statements is always true?

a. AEB is congruent to AED c. ABD is a right triangle.

b. AED is isosceles d. ABC is congruent to CDA

4. Let p represent “The diagonals are congruent,” and let q represent “The diagonals are

perpendicular.” For what quadrilateral is p q true?

a. rhombus c. parallelogram

b. rectangle d. square

5. If quadrilateral ABCD is a parallelogram, which one of the following statements must be

true?

a. AC BD c. AC BD

b. AC and BD bisect each other d. AC bisects DAB and BCD

36

T

M A

H

S

E

D C

A B

6. In the accompanying diagram of parallelogram MATH, 100m T and SH bisects

MHT . What is m HSA?

a. 80 c. 120

b. 140 d. 100

7. In the accompanying diagram of an isosceles trapezoid, AB DC and the diagonals

intersect at E. Which of the following statements is not true?

a. AC BD c. CBA DAB

b. CBD DBA d. ADC ABC

Part II: Short Answer. Show all work!

8. In parallelogram ABCD, 4 17m A x and 2 5m C x . Find the value of x.

37

B

D C

A

DA

B C

9. In the accompanying diagram of parallelogram ABCD, 2 10m A x and

5 15m B x . Find x.

10. In rectangle ABCD, diagonals AC and BD intersect at point E. If

AE = 20 and BD = 2x+30, find x.

11. In rhombus ABCD, the measure of A is 30 more than twice the measure of B . Find

the measure of B ,

12. In the accompanying figure, ABCD is a square, AB = 5x-10 and BC = 2x+20. Find the value

of x.

38

D

A

B

C

BA

D

E

C

D C

BAE

13. In the accompanying figure, ABCD is a rhombus, the lengths of sides AB and BC are

represented by 3x-4 and 2x+1, respectively. Find the value of x.

14. In the accompanying diagram of parallelogram ABCD, side AD is extended through D to

E and DB is a diagonal. If 65m EDC and 85m CBD , find m CDB .

15. In the accompanying diagram, ABCD is a parallelogram, DA DE , and 70m B . Find

m E .

39

N

LM

J K

16. The perimeter of a parallelogram is 32 meters and the two shorter sides each measure 4

meters. What is the length, in meters, of each of the longer sides?

17. In rhombus JKLM, m JML 45 , find

m JNM = ______________

m MJK = ______________

m JKL = _______________

Part III: PROOFS. Show all work!

18. Given: Isosceles trapezoid ABCD with AB parallel to CD ,

A B , and E and F are midpoints of AB and CD , respectively.

Prove: a. ED EC

b. EF CD

40

F

E

CD

A B

19. Given: Rectangle ABCD with

diagonal BFED , AE BD and

CF BD

Prove: AE CF

___________________________________________________________________

20. Given: parallelogram DEBK, BC DA , and DJ BL

Prove: CJ AL

________________________________________________________________________

21. Given: rectangle ABCD, , , ,BNPC AEP DEN and AP DN

Prove: a. ABP DCN

b. AE DE

22. Given: Parallelogram DCTV

BC VS

Prove: 1 2

41

E

F

B

D C

A

G

23. Given:

;AD BC AD BC

Prove: EF FG

_____________________________________________________________________

24.

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