Name _____________________________________
Advanced Geometry/Algebra 2
Assignments for Unit 4 - Chapter 5: Quadrilaterals
*All problems refer to the WRITTEN EXERCISES*
Class
Date
Class Activity
Homework Assignment
Due
Date
Complete classwork
Read section 5.1
Do p. 190 #1, 2, 7, 11, 13, 16, 19, 23, 27, 29.
Complete classwork
Read section 5.2
Section 5.2
Do p. 195 #1, 3, 5, 9, 11, 14, 17, 18, 20.
Proving a Quadrilateral
Also, draw two different quadrilaterals, using a ruler. In
is a Parallelogram
each quadrilateral, join the consecutive midpoints of its
sides to form a new quadrilateral. What kind of a
quadrilateral do you get? Measure in cm!
• Complete classwork
Section 5.3
• Read section 5.3
Special Parallelograms
• Do p. 198 #3, 7, 8, 13, 15, 17, 22, 24, 27, 30, 33.
• Complete classwork
• Read section 5.4
• Do p. 203 ORAL EXERCISES #1;
Section 5.4
• p. 204 #7, 11, 15, 17, 20, 23, 24, 28, 29.
Triangles and Parallel
• Also, given a quadrilateral, suppose you join the
Lines
consecutive midpoints of its sides to form a new
quadrilateral. Prove that this new quadrilateral is a
parallelogram. Measure in cm!
• Complete classwork
Section 5.5
• Read section 5.5
Trapezoids
• Do p. 208 #5, 7, 9, 11, 14, 15, 19–21.
• Chapter 5 review sheet (See solutions on Canvas)
Review Day
• Study by redoing problems!
• Study the reasons from Chapter 3 and 4 too!
We will start Unit 5 (Chapter 8: Similarity) BEFORE the Unit 4 (Ch5 Quadrilaterals) Opportunity
Wed
11/28
Section 5.1
Parallelograms
•
•
•
•
•
•
•
Chapter 5 Opportunity
•
No School Tuesday 11/6 and Monday 11/12
Quarter 1 Closes Friday 11/9
Teachers must complete grades by Sunday 11/18
Report Cards issued in Advisory Tuesday 11/20
No Homework over Thanksgiving Weekend 11/21-11/25
Definitions, Theorems, Postulates, Corollaries, and Properties
used in Geometry Proofs (continued)
Add to this list as needed – Must be a reason in the text or proven/given in class.
Quadrilaterals:
• Definition of a parallelogram (   opp. sides ||)
• If a quad. is a parallelogram, then opposite sides are ≅ . (   opp. sides ≅ )
• If a quad. is a parallelogram, then opposite angles are ≅ . (   opp. ∠ ’s ≅ )
• If a quad. is a parallelogram, then diagonals bisect each other. (   diags bisect each other)
• If opp. sides of a quad. are ≅ , then quad. is a parallelogram. (opp. sides ≅   )
• If opp. angles of a quad. are ≅ , then quad. is a parallelogram. (opp. ∠ ’s ≅   )
• If diagonals of a quad. bisect each other, then quad. is a parallelogram. (diags bisect each other 
)
• If one pair of opposite sides of a quad. are both || and ≅ , then quad. is a parallelogram.
• Diagonals of a rhombus bisect opposite angles.
• Diagonals of a rhombus are ⊥ .
• Diagonals of a rectangle are ≅ .
• Base angles of an isosceles trapezoid are ≅ .
• Diagonals of an isosceles trapezoid are ≅ .
• Diagonals of a kite are ⊥ .
1
5.1 Parallelograms (Day 1)
Definition of parallelogram:
5 Properties of Parallelograms:
1.
(def)
2.
3.
4.
5.
Apply these properties to find values for x, y, and z. Figures a – d are all parallelograms.
a.
b.
y
z
108
29
x
y
10
z
x
120
70
2
c.
d.
y+3
y+1
x–2
x+2
y–6
3y + 1
x–2
2x – 1
Prove the following using the properties of parallelograms.
B
S
Q
E
1
O
G
e.
2
R

L
U
E
N
Given: BENG is a
; SB ≅ NR
Prove: BN and SR bisect each other
N
W
f.

I
Given: QUIN is a
Prove: LU ≅ WN
3
M
E
X
3
U
1
A
K
g. Given: MIKE is a
Prove: ES  UI
 ; ∠1 ≅ ∠2
(From p. 192 #30)
i. Given: ABCD and DEBF are
Prove: AX ≅ CY
D
E
2
5
Y
6
4
C
Z
h. Given: XYZA and XYCZ are
Prove: ∠3 ≅ ∠5
 ’s
 ’s
C
Y
X
A
I
2
S
B
1
F
B
END
4
5.2 Proving that a Quadrilateral is a Parallelogram (Day 2)
Throughout this class you will be proving theorems that allow you to prove that a quadrilateral is a
parallelogram. Each time you prove a new theorem, write the theorem in the box below.
1.
2.
3.
4.

1. If THEO is a
, then its opposite sides are ≅ . Is the converse true? If so, prove it. If not, give a
counterexample.
H
T
E
O
2. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Given: LC and UY bisect each other
L
Prove: LUCY is a
U

Y
C
5
3. If the opposite angles of a quadrilateral are congruent, then it is a parallelogram.
Given:
A
Prove:
K
T
E
4. If one pair of sides are both congruent and parallel, then it is a parallelogram.
Given:
Prove:
A
D
V
I
What if 𝐴𝑉 ≅ 𝐷𝐼 and 𝐴𝐷 ∥ 𝑉𝐼? Does it necessarily follow that DAVI is a parallelogram?
EXPLAIN why or why not.
6
5. In the diagram below, DNIL and DAIE are p-grams.
a. Explain why the diagonals 𝑁𝐿, 𝐼𝐷, and 𝐴𝐸 must all meet at a single point.
A
D
N
b. Prove (2 columns please) that LANE is a p-gram.
L
I
E
Do p. 194 #2, 4, 10, 13, 21 (write steps carefully)
END
7
5.3 Special Parallelograms (Day 3)
Type of p-gram
Definition
Why a parallelogram?
1. How do the diagonals of a rectangle relate to each other?
Rect à
PROVE IT!
2. What conjectures can you make about the diagonals of a rhombus?
Rhombus à
Rhombus à
PROVE THEM!
8
3.a. Explain why a parallelogram with one right angle must have four right angles.
b. Explain why if two consecutive sides of a parallelogram are congruent, then all four sides must be
congruent.
4. Given: MAXH is a parallelogram; MH ≅ HL ; ∠1 ≅ ∠2
Prove: MAXH is a rhombus
(Hint: Use what you just explained in #3)
A
X
1
2
M
H
5. Given: ∠1 ≅ ∠2 ; ∠3 ≅ ∠2 ; JB ≅ BN
Prove: JONB is a rhombus
N
O
4
J
L
1
2 3
B
6. Always/Sometimes/Never: If the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a
rhombus.
9
7. JORD is a quadrilateral such that JO = y + 2, OR = 2y – 1, RD = 3y – 2, JD = y + 1, JR = y + 3, and
OD = 3y – 2. Is there a value of y such that JORD is a rectangle? Explain your reasoning.
A
1 2
8. Given: AE bisects ∠JAL ; JL bisects ∠AJE ; AL  JE
Prove: JALE is a rhombus
6
L
N
4
J
5
3
E
10
Examine the following diagram which shows the relationships between several geometric figures.
POLYGONS
Pentagons
Triangles
Quadrilaterals
Trapezoids
Parallelograms
Rhombi
Squares
Rectangles
Complete each statement below using vocabulary from the Venn diagram. There are many
possible correct statements.
1.
All
are
.
2.
All
are
.
3.
Some
are
.
4.
Some
are
.
5.
No
are
.
6.
No
are
.
END
11
5.4 Triangles and Parallel Lines (Day 4)
Definition of Midsegment:
Triangle Midsegment Theorem: The midsegment of a triangle is
the third side and
its length.
to
Given: D is the midpoint of AC (extend DE to F so DE=EF)
A
E is the midpoint of AB
DE is the midsegment of ∆ABC
Prove: 𝐷𝐸 ∥ 𝐵𝐶
1
DE = BC
2
D
E
C
F
B
1. 𝐴𝐷 ∥ 𝐵𝐸 ∥ 𝐶𝐹; AB ≅ BC
(a) What can you conclude about DE and EF?
Note: You will prove this in tonight’s homework.
and bisects one side à
(p. 203)
(b) If AB = 5x – 7, BC = 3x + 9, DE = 2x + y, and EF = 6y + 1, find x and y.
12
2.
In right triangle ABC below, points D, E, and F are midpoints of the sides.
Furthermore, AB = 26 and AC = 24.
a) Find BC.
b) Find the perimeter of ∆DEF.
3. ∆ABC has perimeter 28 cm. Find the perimeter of the triangle formed by joining the midpoints of the
sides of ∆ABC.
L
4. Points J and E are the midpoints of AX and LX . Find x and y.
5x – y
A
3x+y
9
J
E
14
X
5. Given: A is the midpoint of OX; AB || XY; BC || YZ
Prove: AC || XZ
Hint: Start with ∆OXY and go around counter-clockwise.
6. In ∆ABC, points D, E, and F are midpoints of the sides.
Explain why the four small triangles are congruent to each other.
13
7. This exercise proves that the three medians of any triangle intersect in a point. (From p. 205 #30)
Given: AM and BN are medians of ∆ABC, and O is the point where they intersect;

B
CO intersects AB at P
P
M
O
Follow the steps below to
A
C
N
Prove: CP is a median of ∆ABC
D

On the ray opposite to NB let D be the point such that DN ≅ ON . Draw AD and DC .
a) Given this information and the given extra line segments, prove that ADCO is a parallelogram.
b) Then prove that O is the midpoint of BD .
c) Then prove that P is the midpoint of AB , and hence CP is a median of ∆ABC.
END
14
5.5 Trapezoids (Day 5)
Trapezoid:
Median (Midsegment) of a Trapezoid:
Part A
Angles
1. Plot the following vertices of trapezoid JACO:
J(-8, -4) A(-6, 6) C(6, 6) O(8, -4)
2. Use a protractor to measure the angles.
m∠J =
m∠A =
m∠C =
m ∠ O=
3. What type of trapezoid is this?
4. What can you conclude about the base angles in an isosceles trapezoid?
Part B
Diagonals
5. Draw in the diagonals of trapezoid JACO.
6. Measure the diagonals in centimeters.
JC =
AO =
7. What can you conclude about the diagonals in an isosceles trapezoid?
15
Part C
8. Plot the following vertices of trapezoid GVID
G(-6, -8) V(-7, -3) I(-1, 1) D(6, 0)
9. Plot the midpoint of GV(
,
) and the
midpoint of ID ( , ) and label these
points O and N, respectively.
10. Draw in the line ON. What is this line called?
11. Calculate the slopes:
Slope of VI =
Slope of ON =
Slope of GD =
12. What can you conclude about the midsegment based on the slopes? Explain.
Part D
13. Measure the length of the bases and the midsegment in centimeters.
VI =
ON =
GD =
14. How is the length of the midsegment related to the length of the bases?
Trapezoid Median (Midsegment) Theorem:
16
Properties of Quadrilaterals
Determine if each quadrilateral must have the given property. Check the appropriate boxes for
yes. Leave the box empty for no.
Property
Parallelogram
Rectangle Rhombus Square
Isosceles
Trapezoid
Opposite sides are ||
Opposite sides are ≅
Opposite ∠ ’s are ≅
Consecutive ∠ ’s are
supplementary
Consecutive ∠ ’s are ≅
All ∠ ’s are ≅
All sides are ≅
Diagonals bisect each other
Diagonals are ≅
Diagonals are ⊥
Diagonals bisect of pair of
opposite ∠ ’s
All angles are right ∠ ’s
17
(Day 5 Classwork continued)
1. Prove the following properties of isosceles trapezoids.
(a) Base angles of an isosceles trapezoid are congruent.
Given: Isosceles trapezoid ABCD with BC  AD and AB ≅ CD .
Prove: ∠A ≅ ∠D and ∠B ≅ ∠C
Hint: Draw BE parallel to CD.
(b) The diagonals of an isosceles trapezoid are congruent.
Given: Isosceles trapezoid ABCD with BC  AD and AB ≅ CD .
Prove: AC ≅ BD
2. Shown below is a trapezoid ABCD with AB  CD and midsegment EF .
Find the value of x if:
(a) AB = 21, CD = 4x – 11, and EF = 3x.
(b) AB = 3x + 2, CD = 2x + 1, and EF = 2x + 4
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3. Do p. 209 #20
4. Do p. 208 #16
5. Prove that the segments joining the consecutive midpoints of the sides of an isosceles trapezoid form
a rhombus. (DYOD)
Hint: You already know that it’s a parallelogram.
19
6. When the midpoints of the sides of ABCD are joined, rectangle PQRS is formed.
(a) Draw other quadrilaterals with this property.
(b) What must be true of quadrilateral ABCD if PQRS is to be a rectangle?
Hint: It’s something about the diagonals AC and BD.
(c) Given: P, Q, R, and S are midpoints of the sides of ABCD.
AC ⊥ BD
Prove: PQRS is a rectangle
Hint: You already know that it’s a parallelogram.
7. In quadrilateral ABCD, E is the midpoint of AD , BC = CD, BE = AE, m∠A = 46°, and m∠C =
84°. Find m∠D.
END
20