Download common core 6.2 homework answers

6-2
Properties of Parallelograms
Focus on Reasoning
Essential question: What can you conclude about the sides, angles, and diagonals
of a parallelogram?
TEACH
Standards for
Mathematical Content
1
G-CO.3.11 Prove theorems about parallelograms.
G-SRT.2.5 Use congruence ... criteria for triangles
to solve problems and to prove relationships in
geometric figures.
Materials: geometry software
Questioning Strategies
• As you use the software to drag points
A, B, C, and/or D, does the quadrilateral
remain a parallelogram? Why? Yes; the lines
Vocabulary
diagonal
that form opposite sides remain parallel.
• What do you notice about consecutive angles
in the parallelogram? Why does this make
sense? Consecutive angles are supplementary.
Prerequisites
Theorems about parallel lines cut by a transversal
Triangle congruence criteria
This makes sense because opposite sides are
parallel, so consecutive angles are same-side
interior angles. By the Same-Side Interior Angles
Postulate, these angles are supplementary.
Math Background
In this lesson, students extend their earlier work
with triangle congruence criteria and triangle
properties to prove facts about parallelograms. This
lesson gives students a chance to use inductive and
deductive reasoning to investigate properties of the
sides, angles, and diagonals of parallelograms.
Students have encountered parallelograms
in earlier grades. Ask a volunteer to define
parallelogram. Students may have only an informal
idea of what a parallelogram is (e.g., “a slanted
rectangle”), so be sure they understand that the
mathematical definition of a parallelogram is a
quadrilateral with two pairs of parallel sides. You
may want to show students how they can make
a parallelogram by drawing lines on either side
of a ruler, changing the position of the ruler, and
drawing another pair of lines. Ask students to
explain why this method creates a parallelogram.
2
Prove that opposite sides of a
parallelogram are congruent.
Questioning Strategies
• Why do you think
the proof is based on drawing
___
the diagonal ​DB​ ? Drawing the diagonal
creates two triangles; then you can use triangle
congruence criteria and CPCTC.
245
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
Teaching Strategy
Some students may have difficulty with terms
like opposite sides or consecutive angles. Remind
students that opposite sides of a quadrilateral do
not share a vertex (that is, they do not intersect).
Consecutive sides of a quadrilateral do share a
vertex (that is, they intersect). Opposite angles of
a quadrilateral do not share a side. Consecutive
angles of a quadrilateral do share a side. You
may want to help students draw and label a
quadrilateral for reference.
INTR O D U C E
Chapter 6 Investigate parallelograms.
Name
Class
Notes
6-2
Date
Properties of Parallelograms
Focus on Reasoning
Essential question: What can you conclude about the sides, angles, and diagonals
of a parallelogram?
Recall that a parallelogram is a quadrilateral that
has two pairs of parallel sides. You use the
symbol to name a parallelogram. For example,
the figure shows ABCD.
A
D
1
G-CO.3.11,
G-SRT.2.5
B
C
© Houghton Mifflin Harcourt Publishing Company
Investigate parallelograms.
A
Use the straightedge tool of your geometry
software to draw a straight line. Then plot
a point that is not on the line. Select the
point and line, go to the Construct menu,
and construct a line through the point that
is parallel to the line. This will give you a
pair of parallel lines, as shown.
B
Repeat Step A to construct a second pair
of parallel lines that intersect those from
Step A.
C
The intersections of the parallel lines
create a parallelogram. Plot points
at these intersections. Label the
points A, B, C, and D.
D
Use the Measure menu to measure each
angle of the parallelogram.
E
Use the Measure menu to measure the
length of each side of the parallelogram.
(You can do this by measuring the
distance between consecutive vertices.)
F
B
A
AB = 2.60 cm
BC = 1.74 cm
CD = 2.60 cm
DA = 1.74 cm
C
D
m∠DAB = 116.72˚
m∠ABC = 63.28˚
m∠BCD = 116.72˚
m∠CDA = 63.28˚
Drag the points and lines in your
construction to change the shape of the
parallelogram. As you do so, look for
relationships in the measurements.
REFLECT
1a. Make a conjecture about the sides and angles of a parallelogram.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
245
Chapter 6
Lesson 2
G_MFLBESE200852_C06L02.indd 245
03/05/14 4:56 PM
You may have discovered the following theorem about parallelograms.
Theorem
2
Prove that opposite sides of a parallelogram
are congruent.
A
B
Complete the proof.
Given: ABCD is a parallelogram.
___
____
___
D
___
Prove: AB
≅ CD and AD ≅ BC
Statements
1. ABCD is a parallelogram.
___
___
Reasons
1.
Given
2. Through any two points there exists exactly
one line.
2. Draw DB​.
___
C
___
___
3. AB​​ǁ DC​​;​​AD​​
​ǁ​​BC​​
3.
Definition of parallelogram
4. ∠ADB​≅​∠CBD;​∠ABD​≅​∠CDB
4.
Alternate Interior Angles Theorem
5.
Reflexive Property of Congruence
___
___
5. DB​​≅​​DB​​
6.
△ABD ≅ △CDB
7. AB​≅​CD;​AD​≅​BC
6. ASA Congruence Criterion
7.
CPCTC
REFLECT
2a. Explain how you can use the rotational symmetry of a parallelogram to give an
argument that supports the above theorem.
Under a 180° rotation about the center of the parallelogram, each side is mapped
to its opposite side. Since rotations preserve distance, this shows that opposite
sides are congruent.
2b. One side of a parallelogram is twice as long as another side. The perimeter of
the parallelogram is 24 inches. Is it possible to find all the side lengths of the
parallelogram? If so, find the lengths. If not, explain why not.
Yes; consecutive sides have lengths x, 2x, x, and 2x, so x + 2x + x + 2x = 24, or
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
If a quadrilateral is a parallelogram, then opposite sides are congruent.
6x = 24. Therefore x = 4 and the side lengths are 4 in., 8 in., 4 in., and 8 in.
Chapter 6
G_MFLESE200852_C06L02.indd 246
Chapter 6 246
Lesson 2
08/03/13 10:09 PM
246
Lesson 2
T EACH
3
Highlighting the
Standards
Investigate diagonals of
parallelograms. Materials: geometry software
As students work on the proof in this lesson,
ask them to think about how the format of
the proof makes it easier to understand the
underlying structure of the argument. This
addresses elements of Standard 3 (Construct
viable arguments and critique the reasoning
of others). Students should recognize that a
flow proof shows how one statement connects
to the next. This may not be as apparent in a
two-column format. You may want to have
students rewrite the proof in a two-column
format as a way of exploring this further.
Questioning Strategies
• How many diagonals does a parallelogram have?
Is this true for every quadrilateral? Two; yes
• If a quadrilateral
is ___
named PQRS, what are the
___
diagonals? PR and QS
• Are the diagonals of a parallelogram ever
congruent? If so, when does this appear to happen?
Yes; when the parallelogram is a rectangle
4
Prove diagonals of a parallelogram
bisect each other.
Questioning Strategies
• Why do you think this theorem was introduced
after the theorems about the sides and angles
of a parallelogram? The proof of this theorem
depends upon the fact that opposite sides of
a parallelogram are congruent.
© Houghton Mifflin Harcourt Publishing Company
Chapter 6
247
Lesson 2
Notes
Essential question: What can you conclude about the diagonals of a parallelogram?
A segment that connects any two nonconsecutive
vertices of a polygon is a diagonal. A
parallelogram
___
___ has two diagonals. In the figure,
AC and BD are diagonals of ABCD.
3
A
B
D
C
IInvestigate diagonals of parallelograms.
Use geometry software to construct a
parallelogram. (See Lesson 4-2 for detailed
instructions.) Label the vertices of the
parallelogram A, B, C, and D.
A
B
Use the segment
___ tool
___to construct the
diagonals, AC and BD .
C
Plot a point at the intersection of the
diagonals. Label this point E.
D
Use the Measure
the
___ to measure
___ ___ menu
___
length of AE , BE , CE , and DE . (You can
do this by measuring the distance between
the relevant endpoints.)
A
B
D
C
B
A
AE = 1.52 cm
BE = 2.60 cm
CE = 1.52 cm
DE = 2.60 cm
E
Drag the points and lines in your
construction to change the shape of the
parallelogram. As you do so, look for
relationships in the measurements.
© Houghton Mifflin Harcourt Publishing Company
E
C
D
REFLECT
3a. Make a conjecture about the diagonals of a parallelogram.
The diagonals of a parallelogram bisect each other.
3b. A student claims that the perimeter of AEB is always equal to the perimeter of
CED. Without doing any further measurements in your construction, explain
whether or not you agree with the student’s statement.
Agree; AE = CE, BE = DE, and AB = DC since opposite sides of a parallelogram
are congruent. So, AE + BE + AB = CE + DE + DC.
Chapter 6
Lesson 2
247
You may have discovered the following theorem about parallelograms.
Theorem
T
4
Prove diagonals of a parallelogram bisect each other.
Complete the proof.
C
A
B
Given: ABCD is a parallelogram.
___
___
___
E
___
Prove: AE CE and BE DE .
C
D
ABCD is a
parallelogram.
Given
___ ___
___
___
AB DC
AB DC
Opposite sides of a
parallelogram are congruent.
Definition of
parallelogram
∠ABE ∠CDE
∠BAE ∠DCE
Alternate Interior
Angles Theorem
Alternate Interior
Angles Theorem
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
If a quadrilateral is a parallelogram, then the diagonals bisect each other.
ABE CDE
ASA Congruence Criterion
___
___
___
___
AE CE and BE DE .
CPCTC
REFLECT
4a. Explain how you can prove the theorem using a different congruence criterion.
∠AEB ∠CED because they are vertical angles. Using this fact plus the fact
___
___
that ∠ABE ∠CDE and AB DC , it is possible to prove the theorem using
the AAS Congruence Criterion.
Chapter 6
Chapter 6
248
Lesson 2
248
Lesson 2
CLOS E
Teaching Strategy
The lesson concludes with the theorem that
states that opposite angles of a parallelogram are
congruent. The proof of this theorem is left as an
exercise (Exercise 1). Be sure students recognize
that the proof of this theorem is similar to the
proof that opposite sides of a parallelogram
are congruent. Noticing such similarities is an
important problem-solving skill.
Essential Question
What can you conclude about the sides, angles, and
diagonals of a parallelogram?
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
The diagonals of a parallelogram bisect each other.
Summarize
Have students make a graphic organizer to
summarize what they know about the sides, angles,
and diagonals of a parallelogram. A sample is
shown below.
Highlighting the
Standards
PR ACTICE
Exercise 4 is a multi-part exercise that includes
opportunities for mathematical modeling,
reasoning, and communication. It is a good
opportunity to address Standard 4 (Model with
mathematics). Draw students’ attention to the
way they interpret their mathematical results
in the context of the real-world situation.
Specifically, ask students to explain what their
mathematical findings tell them about the
appearance and layout of the park.
Exercise 1: Students practice what they learned
in part 2 of the lesson.
Exercise 2: Students use reasoning to extend what
they know about parallelograms.
Exercise 3: Students use reasoning and/or algebra
to find unknown angle measures.
Exercise 4: Students apply their learning to solve
a multi-step real-world problem.
Parallelogram
A
B
© Houghton Mifflin Harcourt Publishing Company
C
D
Opposite sides
are congruent.
Opposite angles
are congruent.
AB DC
AD BC
∠ A ∠C
∠ B ∠D
The diagonals
bisect each other.
If E is the point
where diagonals
AC and BD intersect,
then AE CE and
BE DE .
Chapter 6
249
Lesson 2
Notes
The angles of a parallelogram also have an important property. It is stated in the
following theorem, which you will prove as an exercise.
Theorem
T
If a quadrilateral is a parallelogram, then opposite angles are congruent.
PRACTICE
1. Prove the above theorem about opposite angles of a parallelogram.
Given: ABCD is a parallelogram.
Prove: ∠A ∠C and ∠B ∠D
A
(Hint: You only need to prove that ∠A ∠C. A similar
argument can be used to prove that ∠B ∠D. Also,
you may or may not need to use all the rows of the table
in your proof.)
D
C
Statements
Reasons
1.
ABCD is a parallelogram.
2.
Draw DB.
1.
Given
2.
Through any two points there
exists exactly one line.
AB || DC; AD || BC
3.
Definition of parallelogram
∠ADB ∠CBD; ∠ABD ∠CDB
4.
Alternate Interior Angles Theorem
DB DB
5.
Reflexive Property of Congruence
6.
ABD CDB
6.
ASA Congruence Criterion
7.
∠A ∠C
7.
CPCTC
3.
4.
5.
© Houghton Mifflin Harcourt Publishing Company
B
___
___ ___
___
___
___
___
2. Explain why consecutive angles of a parallelogram are supplementary.
Consecutive angles of a parallelogram are same-side interior angles for a
pair of parallel lines (the opposite sides of the parallelogram), so the angles
are supplementary by the Same-Side Interior Angles Postulate.
3. In the figure, JKLM is a parallelogram. Find the measure of each of the
numbered angles.
J
m∠1 = 19°; m∠2 = 43°; m∠3 = 118°;
K
1
3
m∠4 = 118°; m∠5 = 19°
4
N
62˚
43˚
2
5
M
Chapter 6
L
Lesson 2
249
4. A city planner is designing a park in the shape of a
parallelogram. As shown in the figure, there will be
two straight paths through which visitors may enter
the park. The paths are bisectors of consecutive angles
of the parallelogram, and the paths intersect at point P.
A
B
P
D
C
© Houghton Mifflin Harcourt Publishing Company
a. Work directly on the parallelograms below and use a compass and
straightedge to construct the bisectors of ∠A and ∠B. Then use a protractor to
measure ∠APB in each case.
A
B
A
B
P
P
D
C
D
C
Make a conjecture about ∠APB.
∠APB is a right angle.
b. Write a paragraph proof to show that your conjecture is always true. (Hint:
Suppose m∠BAP = x°, m∠ABP = y°, and m∠APB = z°. What do you know
about x + y + z? What do you know about 2x + 2y?)
By the Triangle Sum Theorem, x + y + z = 180. Also, m∠DAB = (2x)° and
m∠ABC = (2y)°. By the Same-Side Interior Angles Postulate m∠DAB +
m∠ABC = 180°. So 2x + 2y = 180 and x + y = 90. Substituting this in the
first equation gives 90 + z = 180 and z = 90.
When the city planner takes into account the
dimensions of the park, she finds that point P lies on
___
DC , as shown. Explain why it must be the case that
DC = 2AD. (Hint: Use congruent base angles to show
that DAP and CPB are isosceles.)
___
A
x˚
B
y˚
x˚
y˚
z˚
D
P
C
∠DAP ∠BAP since AP is an angle bisector. Also, ∠DPA ∠BAP by the
Alternate Interior Angles Theorem. Therefore, ∠DAP ∠DPA. This means
___
___
___
___
___
___
DAP is isosceles, with AD DP . Similarly, BC PC . Also, BC AD as
opposite sides of a parallelogram. So, DC = DP + PC = AD + BC = AD +
© Houghton Mifflin Harcourt Publishing Company
c.
AD = 2AD.
Chapter 6
Chapter 6
250
Lesson 2
250
Lesson 2
ADD I T I O NA L P R AC TI C E
AND P RO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. 108.8 cm
2. 91 cm
3. 217.6 cm
4. 123°
5. 123°
6. 57°
7. 117°
8. 63°
9. 71
10. 21
11. 10.5
12. 15
13. 30
14. (0, -3)
15. Possible answer:
Statements
Reasons
1. Given
2. m∠EDG = m∠EDH +
m∠GDH, m∠FGD =
m∠FGH + m∠DGH
2. Angle Add. Post.
3. m∠EDG + m∠FGD = 180°
3. cons. ∠s supp.
© Houghton Mifflin Harcourt Publishing Company
1. DEFG is a parallelogram.
4. m∠EDH + m∠GDH +
4. Subst. (Steps
m∠FGH + m∠DGH = 180°
2, 3)
5. m∠GDH + m∠DGH +
m∠DHG = 180°
5. Triangle Sum
Thm.
6. m∠GDH + m∠DGH +
m∠DHG = m∠EDH +
m∠GDH + m∠FGH +
m∠DGH
6. Trans. Prop. of =
7. m∠DHG = m∠EDH +
m∠FGH
7. Subtr. Prop. of =
Problem Solving
1. m∠C = 135°; m∠D = 45°
2. 15 in.
3. 4.5 ft
4. 65°
5. B
6. H
7. D
Chapter 6
251
Lesson 2
Name
Class
Notes
6-2
Date
Additional Practice
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