Download Geometry

Geometry
Chapter 5
Midsegment of a Triangle
Median of a Trapezoid
Quadrilaterals
Proofs
Name: _________________________________________________________________
Geometry
Assignments – Chapter 5
Parallel Line Proofs & Chapter 5- Quadrilaterals
Date
Due
Topics
Section

2-2,

5-1 &
Parallelograms & Their
Properties
Rectangles and Their
Properties
5-4
5-4 &
5-5
Written Exercises
Pg 169-170 #2, 5-12, 19-21, 24 & 25
Algebra in Geometry Worksheet - #3,
5, 7, 8





5-3 &
Assignment
Squares & Their Properties
Rhombuses & Their
Properties
Trapezoids & Their Properties
Isosceles Trapezoid & Their
Properties
Worksheet – “More Quads”
Midsegement of triangle
Pg180 #1-9 odd, 16, 17

Median of trapezoid
Pg 192 [written ex at bottom of page]
#1-9, 13-17 odd
5-5
5-2, 5-4
& 5-5

Using the Quadrilaterals to
Prove Parts or Triangles
Worksheeet
5-2, 5-4

Proving the various
Quadrilaterals
Worksheet


More proofs
Review
Worksheet “ Ch 5 Review of Quads”
& 5-5
5-2, 5-4
& 5-5
1
Chap. 5
Review
Suggested Chapter 5 Review
Questions from your Textbook
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)

Use the suggested Practice as
a Guide, Ask for help if you
have questions
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!
2
Chapter 5: Quadrilaterals
Quadrilateral Properties
Parallelogram:
Property
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
Yes
No
3
Rectangle:
Property
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
Yes
No
4
Rhombus:
Property
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
Yes
No
5
Square:
Property
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
Yes
No
6
Trapezoid:
Property
Only one Pair Opposite Sides Parallel
Non-Parallel Sides Congruent
Base Angles Congruent
Consecutive Angles Supplementary
All Angles Congruent (all right angles)
Diagonals Congruent
Diagonals are Perpendicular
Diagonals Bisect Each Other
Diagonals Bisect the Angles
Yes
No
7
Isosceles Trapezoid:
Property
Only one Pair Opposite Sides Parallel
Non-Parallel Sides Congruent
Base Angles Congruent
Consecutive Angles Supplementary
All Angles Congruent (all right angles)
Diagonals Congruent
Diagonals are Perpendicular
Diagonals Bisect Each Other
Diagonals Bisect the Angles
Yes
No
8
Algebra in Geometry quads WS
Geometry
Name ________________________
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.
9
F. Quadrilaterals with perpendicular diagonals.
___________________________________________________________________
1. Two angles formed by one diagonal of a rectangle and the sides of the rectangle are
x2  6 x  50 and 10 x  8 . Find the measures of the two angles.
2. Two consecutive angles in a parallelogram are represented by x 2  6 x  50 and x 2  6 x  50 .
Find all possible measures for the angles.
2
2
3. Opposite sides in a parallelogram are represented by 12 x  10 x and 2 x  13x  3 . If x is
greater than one, find the lengths of the sides of the parallelogram.
2
4. Two consecutive angles of a parallelogram are represented by 2 x  x  50 and
x2  6 x  100 . If x is an integer, find the measures of the angles.
2
5. Opposite angles in a parallelogram are represented by 25x  5x  4 and 15x . Find the
measures of all angles of the parallelogram.
10
6. Two angles formed by one diagonal of a rectangle and the sides of the rectangle are
x2  2 x  50 and 5x  50 . If the smaller angle is greater than 30, find the measures of the two
angles.
7. Two consecutive angles of a parallelogram are x 2  2 x  150 and 3x  6 . Find the measures
of the angles.
8. Opposite sides in a parallelogram are represented by x 2  45 and 2 x  35 . Find the two
possible lengths of the sides.
2
9. Two consecutive angles of a parallelogram are represented by x  20 x  100 and
x2  9 x  59 . If x is a positive integer, find the measures of the angles.
2
2
10. Opposite angles in a parallelogram are represented by 7 x  17 x  21 and 3x  12 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
10 x2  18x  80 and 11x 2  7 . Find the measures of the two angles.
11
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
W
T
Z
2. WZ = 4x – 3, ZS = 13,
RZ = 17, ZT = 7y + 3
R
S
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,
M
<KLM, and <LMJ.
L
N
J
K
12
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.
R
2
S
Z
19. If m<1 = 3x + 12 and m<2 = 2x – 7,
find x and m<1.
1
W
T
13
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.
A
1 2
25. If m<DAB = 7x + 14 and
D
E
3
B
m<2 = 5x – 5, find x and m<1.
C
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.
14
________________________________________________________________________
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.
15
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.
16
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.
17
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
2
IH and J is the midpoint of FI and K is the midpoint of GH . If JK
2
= 8, FG = x +x-2 and IH = x +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
2
BC , M is the midpoint of AB and N is the midpoint of DC . If AD =
2
x +1, MN = 4x+1, and BC = x +2x+1, find AD, MN, and BC.
Ans: 1. 7
2. -3
3. 3 4. 23/11
5. AD = 10, MN = 13, BC = 16
18
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 AC and BD are perpendicular bisectors of each other.
4.
AB
DC , AB  BC , and AD  AB
_____________________________________________________________
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.
19
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 RO and HM intersect at B. If m<BRH =
2
2
x + 3x + 9 and m<BHR = 2 x - 9 , 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.
20
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
21
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
22
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
23
1. Given:
Prove:
PQRS is a parallelogram;
PS  QT
Q
S
T
S
JKLM is a parallelogram;
JO  OL
JKLM is a parallelogram;
JP  QL
Prove: JL and QP bisect each other
J
P
1
J
K
P
1
O
Q
T
R
4. Given:
Prove: OP  OQ
M
Q
P
R
3. Given:
QR  QT
Prove: S  T
QRT is isosceles
P
PQRS is a parallelogram;
2. Given:
K
O
2
L
M
Q
2
L
24
________________________________________________________________________
5. Given:
AECF is a parallelogram;
FD  BE
Prove: AD  BC
ABCD is a parallelogram;
FD  BE
Prove: AF  EC
A
F
6. Given:
B
D
A
E
C
F
B
D
E
C
_______________________________________________________________________
7. Given:
ACEF is a rhombus;
AC  BC
Prove: 1  2
F
E
1
A
2
C
B
25
8. Given:
XYRS is a rectangle;
9. Given:
M is midpoint of YR
Prove: XM  SM
X
XM  SM
Prove: M is midpoint of YR
Y
X
M
R
S
XYRS is a rectangle;
Y
M
S
R
10. Given: Parallelogram ABCD;
DE  FB
Prove:
EGC  FGA
26
11. Given: Parallelogram FLSH;
LG  FS ; HA  FS
Prove:
LGS  HAF
12. Given: ABCD is a parallelogram;
AJ  CE
Prove: GH  FG
27
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
28
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
29
Quad Proofs Day II – Proving Quads
1. Given: AB
DC; AD
2. Given: P  R; Q  S
BC
Prove: 1  2
Prove: PQ RS
A
2
3
D 1
3. Given:
B
P
4 C
5
Q
R
S
AK  BJ ; BJ  BL;
4. Given:
1  3
JKLM is a parallelogram;
PX  QX
Prove: ABJK is a parallelogram Prove: JPLQ is a parallelogram
J
K
B
A
P
X
Q
1
K
2
J
M
3
L
L
30
5. Given:
RX  RY ;
6. Given:
RYS  TZS
RT XZ
Prove: RTZX is a parallelogram
Prove:
Y
RYS  TZS
Y
S
R
X
T
X
ACEF is a parallelogram;
AC  BC; 1  2
Prove: ACEF is a rhombus
F
S
R
Z
7. Given:
RX  RY ; RT  XZ ;
8. Given:
T
Z
ABCD is a rhombus;
ABX  BAX
Prove: ABCD is a square
A
E
B
1
A
X
2
C
B
D
C
31
9. Given:
1  2; 2  3;
3  4
Prove: PQRS is a rhombus
S
P
10. Given:
2 3
Q
PQ  RS
Prove: PQRS is a rhombus
R
4 1
1  2; 2  3;
S
P
R
4 1
2 3
Q
________________________________________________________________________
11. Given: parallelogram ABFE;
Parallelogram EFCD;
AD  CD
Prove: ABCD is a parallelogram
12.
13.
32
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
D
C
F
A
B
E
33
17.
34
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
35
6. In the accompanying diagram of parallelogram MATH, m T  100 and SH bisects
MHT . What is m HSA ?
M
S
A
H
a. 80
c. 120
b. 140
d. 100
T
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?
D
C
E
B
A
a. AC  BD
b. CBD 
c. CBA  DAB
DBA
d. ADC  ABC
Part II: Short Answer. Show all work!
8. In parallelogram ABCD, m A   4 x  17   and m C   2 x  5  . Find the value of x.
36
9. In the accompanying diagram of parallelogram ABCD, m A   2 x  10   and
C
D
m B   5x  15  . Find x.
B
A
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
the measure of B ,
A is 30 more than twice the measure of
B . Find
12. In the accompanying figure, ABCD is a square, AB = 5x-10 and BC = 2x+20. Find the value
of x.
B
A
C
D
37
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. C
B
D
A
14. In the accompanying diagram of parallelogram ABCD, side AD is extended through D to
E and DB is a diagonal. If m EDC  65 and m CBD  85 , find m CDB .
E
C
D
A
B
15. In the accompanying diagram, ABCD is a parallelogram, DA  DE , and m B  70 . Find
m E.
E
A
B
D
C
38
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?
K
J
17. In rhombus JKLM, m JML  45 , find
N
m JNM = ______________
m MJK = ______________
M
L
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
39
A
B
19. Given: Rectangle ABCD with
F
diagonal BFED , AE  BD and
E
CF  BD
D
C
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
ABP  DCN
Prove: a.
b. AE  DE
22. Given: Parallelogram DCTV
BC VS
Prove:
1
2
40
23. Given:
D
G
AD  BC; AD BC
C
F
Prove: EF  FG
A
E
B
_____________________________________________________________________
24.
41
42