Lesson 28: Properties of Parallelograms

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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 28
M1
GEOMETRY
Lesson 28: Properties of Parallelograms
Student Outcomes
๏‚ง
Students complete proofs that incorporate properties of parallelograms.
Lesson Notes
Throughout this module, we have seen the theme of building new facts with the use of established ones. We see this
again in Lesson 28, where triangle congruence criteria are used to demonstrate why certain properties of parallelograms
hold true. We begin establishing new facts using only the definition of a parallelogram and the properties we have
assumed when proving statements. Students combine the basic definition of a parallelogram with triangle congruence
criteria to yield properties taken for granted in earlier grades, such as opposite sides of a parallelogram are parallel.
Classwork
Opening Exercise (5 minutes)
Opening Exercise
a.
If the triangles are congruent, state the congruence.
โ–ณ ๐‘จ๐‘ฎ๐‘ป ≅ โ–ณ ๐‘ด๐’€๐‘ฑ
b.
Which triangle congruence criterion guarantees part 1?
AAS
c.
ฬ…ฬ…ฬ…ฬ… corresponds with
๐‘ป๐‘ฎ
ฬ…ฬ…ฬ…
๐‘ฑ๐’€
Discussion/Examples 1–7 (35 minutes)
Discussion
How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can
we now prove about the properties of parallelograms?
To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel.
However, we have assumed other details about parallelograms to be true, too. We assume that:
๏‚ง
Opposite sides are congruent.
๏‚ง
Opposite angles are congruent.
๏‚ง
Diagonals bisect each other.
Let us examine why each of these properties is true.
Lesson 28:
Properties of Parallelograms
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This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Example 1
If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure. Complete the diagram, and
develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why opposite
sides and angles of a parallelogram are congruent.
Given:
ฬ…ฬ…ฬ…ฬ…, ๐‘จ๐‘ซ
ฬ…ฬ…ฬ…ฬ…)
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ช๐‘ซ
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ช๐‘ฉ
Parallelogram ๐‘จ๐‘ฉ๐‘ช๐‘ซ (๐‘จ๐‘ฉ
Prove:
๐‘จ๐‘ซ = ๐‘ช๐‘ฉ, ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ, ๐’Ž∠๐‘จ = ๐’Ž∠๐‘ช, ๐’Ž∠๐‘ฉ = ๐’Ž∠๐‘ซ
A
Construction: Label the quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ, and mark opposite sides as
parallel. Draw diagonal ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ซ.
B
PROOF:
Parallelogram ๐‘จ๐‘ฉ๐‘ช๐‘ซ
Given
๐’Ž∠๐‘จ๐‘ฉ๐‘ซ = ๐’Ž∠๐‘ช๐‘ซ๐‘ฉ
If parallel lines are cut by a
transversal, then alternate interior angles are equal in measure.
๐‘ฉ๐‘ซ = ๐‘ซ๐‘ฉ
Reflexive property
๐’Ž∠๐‘ช๐‘ฉ๐‘ซ = ๐’Ž∠๐‘จ๐‘ซ๐‘ฉ
If parallel lines are cut by a transversal, then alternate interior angles
are equal in measure.
โ–ณ ๐‘จ๐‘ฉ๐‘ซ ≅โ–ณ ๐‘ช๐‘ซ๐‘ฉ
ASA
๐‘จ๐‘ซ = ๐‘ช๐‘ฉ, ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ
Corresponding sides of congruent triangles are equal in length.
๐’Ž∠๐‘จ = ๐’Ž∠๐‘ช
Corresponding angles of congruent triangles are equal in measure.
D
C
๐’Ž∠๐‘จ๐‘ฉ๐‘ซ + ๐’Ž∠๐‘ช๐‘ฉ๐‘ซ = ๐’Ž∠๐‘จ๐‘ฉ๐‘ช,
๐’Ž∠๐‘ช๐‘ซ๐‘ฉ + ๐’Ž∠๐‘จ๐‘ซ๐‘ฉ = ๐’Ž∠๐‘จ๐‘ซ๐‘ช
Angle addition postulate
๐’Ž∠๐‘จ๐‘ฉ๐‘ซ + ๐’Ž∠๐‘ช๐‘ฉ๐‘ซ = ๐’Ž∠๐‘ช๐‘ซ๐‘ฉ + ๐’Ž∠๐‘จ๐‘ซ๐‘ฉ
Addition property of equality
๐’Ž∠๐‘ฉ = ๐’Ž∠๐‘ซ
Substitution property of equality
Example 2
If a quadrilateral is a parallelogram, then the diagonals bisect each other. Complete the diagram, and develop an
appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a
parallelogram bisect each other. Remember, now that we have proved opposite sides and angles of a parallelogram to be
congruent, we are free to use these facts as needed (i.e., ๐‘จ๐‘ซ = ๐‘ช๐‘ฉ, ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ, ∠๐‘จ ≅ ∠๐‘ช, ∠๐‘ฉ ≅ ∠๐‘ซ).
Given:
Parallelogram ๐‘จ๐‘ฉ๐‘ช๐‘ซ
Prove:
Diagonals bisect each other, ๐‘จ๐‘ฌ = ๐‘ช๐‘ฌ, ๐‘ซ๐‘ฌ = ๐‘ฉ๐‘ฌ
Construction: Label the quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ. Mark opposite sides as
parallel. Draw diagonals ฬ…ฬ…ฬ…ฬ…
๐‘จ๐‘ช and ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ซ.
PROOF:
Parallelogram ๐‘จ๐‘ฉ๐‘ช๐‘ซ
Given
๐’Ž∠๐‘ฉ๐‘จ๐‘ช = ๐’Ž∠๐‘ซ๐‘ช๐‘จ
If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.
๐’Ž∠๐‘จ๐‘ฌ๐‘ฉ = ๐’Ž∠๐‘ช๐‘ฌ๐‘ซ
Vertical angles are equal in measure.
๐‘จ๐‘ฉ = ๐‘ช๐‘ซ
Opposite sides of a parallelogram are equal in length.
โ–ณ ๐‘จ๐‘ฌ๐‘ฉ ≅ โ–ณ ๐‘ช๐‘ฌ๐‘ซ
AAS
๐‘จ๐‘ฌ = ๐‘ช๐‘ฌ, ๐‘ซ๐‘ฌ = ๐‘ฉ๐‘ฌ
Corresponding sides of congruent triangles are equal in length.
Lesson 28:
Properties of Parallelograms
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Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Now we have established why the properties of parallelograms that we have assumed to be true are in fact true. By
extension, these facts hold for any type of parallelogram, including rectangles, squares, and rhombuses. Let us look at
one last fact concerning rectangles. We established that the diagonals of general parallelograms bisect each other. Let us
now demonstrate that a rectangle has congruent diagonals.
Students may need a reminder that a rectangle is a parallelogram with four right angles.
Example 3
If the parallelogram is a rectangle, then the diagonals are equal in length. Complete the diagram, and develop an
appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a rectangle
are congruent. As in the last proof, remember to use any already proven facts as needed.
Given:
Rectangle ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ
Prove:
Diagonals are equal in length, ๐‘ฎ๐‘ฐ = ๐‘ฏ๐‘ฑ
Construction: Label the rectangle ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ. Mark opposite sides as parallel, and
add small squares at the vertices to indicate ๐Ÿ—๐ŸŽ° angles. Draw
ฬ…ฬ…ฬ…ฬ… and ๐‘ฏ๐‘ฑ
ฬ…ฬ…ฬ…ฬ….
diagonals ๐‘ฎ๐‘ฐ
PROOF:
Rectangle ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ
Given
๐‘ฎ๐‘ฑ = ๐‘ฐ๐‘ฏ
Opposite sides of a parallelogram are equal in length.
๐‘ฎ๐‘ฏ = ๐‘ฏ๐‘ฎ
Reflexive property
∠๐‘ฑ๐‘ฎ๐‘ฏ, ∠๐‘ฐ๐‘ฏ๐‘ฎ are right angles.
Definition of a rectangle
โ–ณ ๐‘ฎ๐‘ฏ๐‘ฑ ≅ โ–ณ ๐‘ฏ๐‘ฎ๐‘ฐ
SAS
๐‘ฎ๐‘ฐ = ๐‘ฏ๐‘ฑ
Corresponding sides of congruent triangles are equal in length.
Converse Properties: Now we examine the converse of each of the properties we proved. Begin with the property, and
prove that the quadrilateral is in fact a parallelogram.
Example 4
If both pairs of opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an
appropriate diagram, and provide the relevant Given and Prove for this case.
Given:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ with ๐’Ž∠๐‘จ = ๐’Ž∠๐‘ช, ๐’Ž∠๐‘ฉ = ๐’Ž∠๐‘ซ
Prove:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
Lesson 28:
Properties of Parallelograms
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Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Construction: Label the quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ. Mark opposite angles as congruent.
ฬ…ฬ…ฬ…ฬ…ฬ…. Label the measures of ∠๐‘จ and ∠๐‘ช as ๐’™°. Label
Draw diagonal ๐‘ฉ๐‘ซ
the measures of the four angles created by ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ซ as ๐’“°, ๐’”°, ๐’•°, and ๐’–°.
A
t
x
o
B
o
u
o
PROOF:
r
o
x
s
o
o
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ with ๐’Ž∠๐‘จ = ๐’Ž∠๐‘ช, ๐’Ž∠๐‘ฉ = ๐’Ž∠๐‘ซ
Given
๐’Ž∠๐‘ซ = ๐’“ + ๐’”, ๐’Ž∠๐‘ฉ = ๐’• + ๐’–
Angle addition
๐’“+๐’” = ๐’•+๐’–
Substitution
๐’™ + ๐’“ + ๐’• = ๐Ÿ๐Ÿ–๐ŸŽ, ๐’™ + ๐’” + ๐’– = ๐Ÿ๐Ÿ–๐ŸŽ
Angles in a triangle add up to ๐Ÿ๐Ÿ–๐ŸŽ°.
๐’“+๐’• =๐’”+๐’–
Subtraction property of equality, substitution
๐’“ + ๐’• − (๐’“ + ๐’”) = ๐’” + ๐’– − (๐’• + ๐’–)
Subtraction property of equality
๐’•−๐’” = ๐’”−๐’•
Additive inverse property
๐’• − ๐’” + (๐’” − ๐’•) = ๐’” − ๐’• + (๐’” − ๐’•)
Addition property of equality
๐ŸŽ = ๐Ÿ(๐’” − ๐’•)
Addition and subtraction properties of equality
๐ŸŽ= ๐’”−๐’•
Division property of equality
๐’”=๐’•
Addition property of equality
๐’”=๐’•⇒๐’“=๐’–
Substitution and subtraction properties of equality
ฬ…ฬ…ฬ…ฬ…
๐‘จ๐‘ฉ โˆฅ ฬ…ฬ…ฬ…ฬ…
๐‘ช๐‘ซ, ฬ…ฬ…ฬ…ฬ…
๐‘จ๐‘ซ โˆฅ ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ช
If two lines are cut by a transversal such that a pair of
alternate interior angles are equal in measure, then the lines
are parallel.
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
Definition of a parallelogram
D
C
Example 5
If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an appropriate
diagram, and provide the relevant Given and Prove for this case.
Given:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ with ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ, ๐‘จ๐‘ซ = ๐‘ฉ๐‘ช
Prove:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
Construction: Label the quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ, and mark opposite sides as
equal. Draw diagonal ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ซ.
A
B
PROOF:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ with ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ, ๐‘จ๐‘ซ = ๐‘ช๐‘ฉ
Given
๐‘ฉ๐‘ซ = ๐‘ซ๐‘ฉ
Reflexive property
โ–ณ ๐‘จ๐‘ฉ๐‘ซ ≅ โ–ณ ๐‘ช๐‘ซ๐‘ฉ
SSS
∠๐‘จ๐‘ฉ๐‘ซ ≅ ∠๐‘ช๐‘ซ๐‘ฉ, ∠๐‘จ๐‘ซ๐‘ฉ ≅ ∠๐‘ช๐‘ฉ๐‘ซ
Corresponding angles of congruent triangles are congruent.
ฬ…ฬ…ฬ…ฬ…, ๐‘จ๐‘ซ
ฬ…ฬ…ฬ…ฬ…
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ช๐‘ซ
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ช๐‘ฉ
๐‘จ๐‘ฉ
If two lines are cut by a transversal and the alternate interior angles
are congruent, then the lines are parallel.
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
Definition of a parallelogram
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
D
C
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Example 6
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Draw an appropriate
diagram, and provide the relevant Given and Prove for this case. Use triangle congruence criteria to demonstrate why the
quadrilateral is a parallelogram.
Given:
ฬ…ฬ…ฬ…ฬ… andฬ…ฬ…ฬ…ฬ…ฬ…
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ, diagonals ๐‘จ๐‘ช
๐‘ฉ๐‘ซ bisect each other.
Prove:
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
A
Construction: Label the quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ, and mark opposite sides as equal.
Draw diagonals ฬ…ฬ…ฬ…ฬ…
๐‘จ๐‘ช and ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ฉ๐‘ซ.
PROOF:
B
E
ฬ…ฬ…ฬ…ฬ… and ๐‘ฉ๐‘ซ
ฬ…ฬ…ฬ…ฬ…ฬ… bisect each other.
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ, diagonals ๐‘จ๐‘ช
Given
๐‘จ๐‘ฌ = ๐‘ช๐‘ฌ, ๐‘ซ๐‘ฌ = ๐‘ฉ๐‘ฌ
Definition of a segment bisector
๐’Ž∠๐‘ซ๐‘ฌ๐‘ช = ๐’Ž∠๐‘ฉ๐‘ฌ๐‘จ, ๐’Ž∠๐‘จ๐‘ฌ๐‘ซ = ๐’Ž∠๐‘ช๐‘ฌ๐‘ฉ
Vertical angles are equal in measure.
โ–ณ ๐‘ซ๐‘ฌ๐‘ช ≅ โ–ณ ๐‘ฉ๐‘ฌ๐‘จ, โ–ณ ๐‘จ๐‘ฌ๐‘ซ ≅ โ–ณ ๐‘ช๐‘ฌ๐‘ฉ
SAS
∠๐‘จ๐‘ฉ๐‘ซ ≅ ∠๐‘ช๐‘ซ๐‘ฉ, ∠๐‘จ๐‘ซ๐‘ฉ ≅ ∠๐‘ช๐‘ฉ๐‘ซ
Corresponding angles of congruent triangles are
congruent.
ฬ…ฬ…ฬ…ฬ… ,ฬ…ฬ…ฬ…ฬ…ฬ…
ฬ…ฬ…ฬ…ฬ…
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ช๐‘ซ
๐‘จ๐‘ฉ
๐‘จ๐‘ซ โˆฅ ๐‘ช๐‘ฉ
If two lines are cut by a transversal such that a pair of
alternate interior angles are congruent, then the lines
are parallel.
Quadrilateral ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram.
Definition of a parallelogram
D
C
Example 7
If the diagonals of a parallelogram are equal in length, then the parallelogram is a rectangle. Complete the diagram, and
develop an appropriate Given and Prove for this case.
Given:
Parallelogram ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ with diagonals of equal length, ๐‘ฎ๐‘ฐ = ๐‘ฏ๐‘ฑ
Prove:
๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ is a rectangle.
ฬ…ฬ…ฬ… and ฬ…ฬ…ฬ…ฬ…
Construction: Label the quadrilateral ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ. Draw diagonals ฬ…๐‘ฎ๐‘ฐ
๐‘ฏ๐‘ฑ.
G
H
J
I
PROOF:
Parallelogram ๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ with diagonals of equal length, ๐‘ฎ๐‘ฐ = ๐‘ฏ๐‘ฑ
Given
๐‘ฎ๐‘ฑ = ๐‘ฑ๐‘ฎ, ๐‘ฏ๐‘ฐ = ๐‘ฐ๐‘ฏ
Reflexive property
๐‘ฎ๐‘ฏ = ๐‘ฐ๐‘ฑ
Opposite sides of a parallelogram are congruent.
โ–ณ ๐‘ฏ๐‘ฑ๐‘ฎ ≅ โ–ณ ๐‘ฐ๐‘ฎ๐‘ฑ, โ–ณ ๐‘ฎ๐‘ฏ๐‘ฐ ≅ โ–ณ ๐‘ฑ๐‘ฐ๐‘ฏ
SSS
๐’Ž∠๐‘ฎ = ๐’Ž∠๐‘ฑ, ๐’Ž∠๐‘ฏ = ๐’Ž∠๐‘ฐ
Corresponding angles of congruent triangles are equal
in measure.
๐’Ž∠๐‘ฎ + ๐’Ž∠๐‘ฑ = ๐Ÿ๐Ÿ–๐ŸŽ°, ๐’Ž∠๐‘ฏ + ๐’Ž∠๐‘ฐ = ๐Ÿ๐Ÿ–๐ŸŽ°
If parallel lines are cut by a transversal, then interior
angles on the same side are supplementary.
๐Ÿ(๐’Ž∠๐‘ฎ) = ๐Ÿ๐Ÿ–๐ŸŽ°, ๐Ÿ(๐’Ž∠๐‘ฏ) = ๐Ÿ๐Ÿ–๐ŸŽ°
Substitution property of equality
๐’Ž∠๐‘ฎ = ๐Ÿ—๐ŸŽ°, ๐’Ž∠๐‘ฏ = ๐Ÿ—๐ŸŽ°
Division property of equality
๐’Ž∠๐‘ฎ = ๐’Ž∠๐‘ฑ = ๐’Ž∠๐‘ฏ = ๐’Ž∠๐‘ฐ = ๐Ÿ—๐ŸŽ°
Substitution property of equality
๐‘ฎ๐‘ฏ๐‘ฐ๐‘ฑ is a rectangle.
Definition of a rectangle
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 28
M1
GEOMETRY
Closing (1 minute)
๏‚ง
A parallelogram is a quadrilateral whose opposite sides are parallel. We have proved the following properties
of a parallelogram to be true: Opposite sides are congruent, opposite angles are congruent, diagonals bisect
each other.
Exit Ticket (5 minutes)
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name ___________________________________________________
Date____________________
Lesson 28: Properties of Parallelograms
Exit Ticket
Given:
Equilateral parallelogram ๐ด๐ต๐ถ๐ท (i.e., a rhombus) with diagonals ฬ…ฬ…ฬ…ฬ…
๐ด๐ถ and ฬ…ฬ…ฬ…ฬ…
๐ต๐ท
Prove:
Diagonals intersect perpendicularly.
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Exit Ticket Sample Solutions
ฬ…ฬ…ฬ…ฬ… and ๐‘ฉ๐‘ซ
ฬ…ฬ…ฬ…ฬ…ฬ…
Given: Equilateral parallelogram ๐‘จ๐‘ฉ๐‘ช๐‘ซ (i.e., a rhombus) with diagonals ๐‘จ๐‘ช
Prove: Diagonals intersect perpendicularly.
Rhombus ๐‘จ๐‘ฉ๐‘ช๐‘ซ
Given
๐‘จ๐‘ฌ = ๐‘ช๐‘ฌ, ๐‘ซ๐‘ฌ = ๐‘ฉ๐‘ฌ
Diagonals of a parallelogram
bisect each other.
๐‘จ๐‘ฉ = ๐‘ฉ๐‘ช = ๐‘ช๐‘ซ = ๐‘ซ๐‘จ
Definition of equilateral
parallelogram
โ–ณ ๐‘จ๐‘ฌ๐‘ซ ≅ โ–ณ ๐‘จ๐‘ฌ๐‘ฉ ≅ โ–ณ ๐‘ช๐‘ฌ๐‘ฉ ≅ โ–ณ ๐‘ช๐‘ฌ๐‘ซ
SSS
๐’Ž∠๐‘จ๐‘ฌ๐‘ซ = ๐’Ž∠๐‘จ๐‘ฌ๐‘ฉ = ๐’Ž∠๐‘ฉ๐‘ฌ๐‘ช = ๐’Ž∠๐‘ช๐‘ฌ๐‘ซ
Corr. angles of congruent triangles are equal in measure.
๐’Ž∠๐‘จ๐‘ฌ๐‘ซ = ๐’Ž∠๐‘จ๐‘ฌ๐‘ฉ = ๐’Ž∠๐‘ฉ๐‘ฌ๐‘ช = ๐’Ž∠๐‘ช๐‘ฌ๐‘ซ = ๐Ÿ—๐ŸŽ°
Angles at a point sum to ๐Ÿ‘๐Ÿ”๐ŸŽ°. Since all four angles are congruent,
each angle measures ๐Ÿ—๐ŸŽ°.
Problem Set Sample Solutions
Use the facts you have established to complete exercises involving different types of parallelograms.
1.
ฬ…ฬ…ฬ…ฬ… โˆฅ ฬ…ฬ…ฬ…ฬ…
Given: ๐‘จ๐‘ฉ
๐‘ช๐‘ซ, ๐‘จ๐‘ซ = ๐‘จ๐‘ฉ, ๐‘ช๐‘ซ = ๐‘ช๐‘ฉ
Prove: ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a rhombus.
ฬ…ฬ…ฬ…ฬ….
Construction: Draw diagonal ๐‘จ๐‘ช
2.
๐‘จ๐‘ซ = ๐‘จ๐‘ฉ, ๐‘ช๐‘ซ = ๐‘ช๐‘ฉ
Given
๐‘จ๐‘ช = ๐‘ช๐‘จ
Reflexive property
โ–ณ ๐‘จ๐‘ซ๐‘ช ≅ โ–ณ ๐‘ช๐‘ฉ๐‘จ
SSS
๐‘จ๐‘ซ = ๐‘ช๐‘ฉ, ๐‘จ๐‘ฉ = ๐‘ช๐‘ซ
Corresponding sides of congruent triangles are equal in length.
๐‘จ๐‘ฉ = ๐‘ฉ๐‘ช = ๐‘ช๐‘ซ = ๐‘จ๐‘ซ
Transitive property
๐‘จ๐‘ฉ๐‘ช๐‘ซ is a rhombus.
Definition of a rhombus
ฬ…ฬ…ฬ…ฬ….
Given: Rectangle ๐‘น๐‘บ๐‘ป๐‘ผ, ๐‘ด is the midpoint of ๐‘น๐‘บ
Prove: โ–ณ ๐‘ผ๐‘ด๐‘ป is isosceles.
Rectangle ๐‘น๐‘บ๐‘ป๐‘ผ
Given
๐‘น๐‘ผ = ๐‘บ๐‘ป
Opposite sides of a rectangle are
congruent.
∠๐‘น, ∠๐‘บ are right angles.
Definition of a rectangle
ฬ…ฬ…ฬ…ฬ….
๐‘ด is the midpoint of ๐‘น๐‘บ
Given
๐‘น๐‘ด = ๐‘บ๐‘ด
Definition of a midpoint
โ–ณ ๐‘น๐‘ด๐‘ผ ≅ โ–ณ ๐‘บ๐‘ด๐‘ป
SAS
ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ผ๐‘ด ≅ ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘ป๐‘ด
Corresponding sides of congruent triangles are congruent.
โ–ณ ๐‘ผ๐‘ด๐‘ป is isosceles.
Definition of an isosceles triangle
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
3.
ฬ…ฬ…ฬ…ฬ… bisects ∠๐‘ช๐‘ฉ๐‘จ.
ฬ…ฬ…ฬ…ฬ…ฬ… bisects ∠๐‘จ๐‘ซ๐‘ช, ๐‘บ๐‘ฉ
Given: ๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram, ๐‘น๐‘ซ
Prove: ๐‘ซ๐‘น๐‘ฉ๐‘บ is a parallelogram.
๐‘จ๐‘ฉ๐‘ช๐‘ซ is a parallelogram;
Given
ฬ…ฬ…ฬ…ฬ…ฬ…
๐‘น๐‘ซ bisects ∠๐‘จ๐‘ซ๐‘ช, ฬ…ฬ…ฬ…ฬ…
๐‘บ๐‘ฉ bisects ∠๐‘ช๐‘ฉ๐‘จ.
๐‘จ๐‘ซ = ๐‘ช๐‘ฉ
Opposite sides of a parallelogram are congruent.
∠๐‘จ ≅ ∠๐‘ช, ∠๐‘ฉ ≅ ∠๐‘ซ
Opposite angles of a parallelogram are congruent.
∠๐‘น๐‘ซ๐‘จ ≅ ∠๐‘น๐‘ซ๐‘บ, ∠๐‘บ๐‘ฉ๐‘ช ≅ ∠๐‘บ๐‘ฉ๐‘น
Definition of angle bisector
๐’Ž∠๐‘น๐‘ซ๐‘จ + ๐’Ž∠๐‘น๐‘ซ๐‘บ = ๐’Ž∠๐‘ซ, ๐’Ž∠๐‘บ๐‘ฉ๐‘ช + ๐’Ž∠๐‘บ๐‘ฉ๐‘น = ๐’Ž∠๐‘ฉ
Angle addition
๐’Ž∠๐‘น๐‘ซ๐‘จ + ๐’Ž∠๐‘น๐‘ซ๐‘จ = ๐’Ž∠๐‘ซ, ๐’Ž∠๐‘บ๐‘ฉ๐‘ช + ๐’Ž∠๐‘บ๐‘ฉ๐‘ช = ๐’Ž∠๐‘ฉ
Substitution
๐Ÿ(๐’Ž∠๐‘น๐‘ซ๐‘จ) = ๐’Ž∠๐‘ซ, ๐Ÿ(๐’Ž∠๐‘บ๐‘ฉ๐‘ช) = ๐’Ž∠๐‘ฉ
Addition
๐Ÿ
๐Ÿ
4.
๐Ÿ
๐Ÿ
๐’Ž∠๐‘น๐‘ซ๐‘จ = ๐’Ž∠๐‘ซ, ๐’Ž∠๐‘บ๐‘ฉ๐‘ช = ๐’Ž∠๐‘ฉ
Division
∠๐‘น๐‘ซ๐‘จ ≅ ∠๐‘บ๐‘ฉ๐‘ช
Substitution
โ–ณ ๐‘ซ๐‘จ๐‘น ≅ โ–ณ ๐‘ฉ๐‘ช๐‘บ
ASA
∠๐‘ซ๐‘น๐‘จ ≅ ∠๐‘ฉ๐‘บ๐‘ช
Corresponding angles of congruent triangles are
congruent.
∠๐‘ซ๐‘น๐‘ฉ ≅ ∠๐‘ฉ๐‘บ๐‘ซ
Supplements of congruent angles are congruent.
๐‘ซ๐‘น๐‘ฉ๐‘บ is a parallelogram.
Opposite angles of quadrilateral ๐‘ซ๐‘น๐‘ฉ๐‘บ are
congruent.
Given: ๐‘ซ๐‘ฌ๐‘ญ๐‘ฎ is a rectangle, ๐‘พ๐‘ฌ = ๐’€๐‘ฎ, ๐‘พ๐‘ฟ = ๐’€๐’
Prove: ๐‘พ๐‘ฟ๐’€๐’ is a parallelogram.
๐‘ซ๐‘ฌ = ๐‘ญ๐‘ฎ, ๐‘ซ๐‘ฎ = ๐‘ญ๐‘ฌ
Opposite sides of a
rectangle are congruent.
๐‘ซ๐‘ฌ๐‘ญ๐‘ฎ is a rectangle;
Given
๐‘พ๐‘ฌ = ๐’€๐‘ฎ, ๐‘พ๐‘ฟ = ๐’€๐’
๐‘ซ๐‘ฌ = ๐‘ซ๐‘พ + ๐‘พ๐‘ฌ, ๐‘ญ๐‘ฎ = ๐’€๐‘ฎ + ๐‘ญ๐’€
Segment addition
๐‘ซ๐‘พ + ๐‘พ๐‘ฌ = ๐’€๐‘ฎ + ๐‘ญ๐’€
Substitution
๐‘ซ๐‘พ + ๐’€๐‘ฎ = ๐’€๐‘ฎ + ๐‘ญ๐’€
Substitution
๐‘ซ๐‘พ = ๐‘ญ๐’€
Subtraction
๐’Ž∠๐‘ซ = ๐’Ž∠๐‘ฌ = ๐’Ž∠๐‘ญ = ๐’Ž∠๐‘ฎ = ๐Ÿ—๐ŸŽ°
Definition of a rectangle
โ–ณ ๐’๐‘ฎ๐’€, โ–ณ ๐‘ฟ๐‘ฌ๐‘พ are right triangles.
Definition of right triangle
โ–ณ ๐’๐‘ฎ๐’€ ≅ โ–ณ ๐‘ฟ๐‘ฌ๐‘พ
HL
๐’๐‘ฎ = ๐‘ฟ๐‘ฌ
Corresponding sides of congruent triangles are congruent.
๐‘ซ๐‘ฎ = ๐’๐‘ฎ + ๐‘ซ๐’; ๐‘ญ๐‘ฌ = ๐‘ฟ๐‘ฌ + ๐‘ญ๐‘ฟ
Partition property or segment addition
๐‘ซ๐’ = ๐‘ญ๐‘ฟ
Subtraction property of equality
โ–ณ ๐‘ซ๐’๐‘พ ≅ โ–ณ ๐‘ญ๐‘ฟ๐’€
SAS
๐’๐‘พ = ๐‘ฟ๐’€
Corresponding sides of congruent triangles are congruent.
๐‘พ๐‘ฟ๐’€๐’ is a parallelogram.
Both pairs of opposite sides of a parallelogram are congruent.
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
241
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 28
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
5.
ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… and ๐‘ซ๐‘ฌ๐‘ญ
ฬ…ฬ…ฬ…ฬ…ฬ…ฬ… are segments.
Given: Parallelogram ๐‘จ๐‘ฉ๐‘ญ๐‘ฌ, ๐‘ช๐‘น = ๐‘ซ๐‘บ, ๐‘จ๐‘ฉ๐‘ช
Prove: ๐‘ฉ๐‘น = ๐‘บ๐‘ฌ
ฬ…ฬ…ฬ…ฬ… โˆฅ ๐‘ญ๐‘ฌ
ฬ…ฬ…ฬ…ฬ…
๐‘จ๐‘ฉ
Opposite sides of a
parallelogram are
parallel.
๐’Ž∠๐‘ฉ๐‘ช๐‘น = ๐’Ž∠๐‘ฌ๐‘ซ๐‘บ
If parallel lines are cut by
a transversal, then
alternate interior angles are equal in measure.
∠๐‘จ๐‘ฉ๐‘ญ ≅ ∠๐‘ญ๐‘ฌ๐‘จ
Opposite angles of a parallelogram are congruent.
∠๐‘ช๐‘ฉ๐‘น ≅ ∠๐‘ซ๐‘ฌ๐‘บ
Supplements of congruent angles are congruent.
๐‘ช๐‘น = ๐‘ซ๐‘บ
Given
โ–ณ ๐‘ช๐‘ฉ๐‘น ≅ โ–ณ ๐‘ซ๐‘ฌ๐‘บ
AAS
๐‘ฉ๐‘น = ๐‘บ๐‘ฌ
Corresponding sides of congruent triangles are equal in length.
Lesson 28:
Properties of Parallelograms
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M1-TE-1.3.0-07.2015
242
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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