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Five-Minute Check (over Lesson 6–2)
CCSS
Then/Now
Theorems: Conditions for Parallelograms
Proof: Theorem 6.9
Example 1: Identify Parallelograms
Example 2: Real-World Example: Use Parallelograms to Prove
Relationships
Example 3: Use Parallelograms and Algebra to Find Values
Concept Summary: Prove that a Quadrilateral Is a
Parallelogram
Example 4: Parallelograms and Coordinate Geometry
Example 5: Parallelograms and Coordinate Proofs
Over Lesson 6–2
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Over Lesson 6–2
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B.
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Over Lesson 6–2
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A. A
B. B
C. C
Over Lesson 6–2
An expandable gate is made of parallelograms that
have angles that change
measure as the gate is
adjusted. Which of the
following statements is
always true?
A. A  C and B  D
B. A  B and C  D
C.
D.
Content Standards
G.CO.11 Prove theorems about
parallelograms.
G.GPE.4 Use coordinates to prove simple
geometric theorems algebraically.
Mathematical Practices
3 Construct viable arguments and critique
the reasoning of others.
2 Reason abstractly and quantitatively.
You recognized and applied properties of
parallelograms.
• Recognize the conditions that ensure a
quadrilateral is a parallelogram.
• Prove that a set of points forms a
parallelogram in the coordinate plane.
Identify Parallelograms
Determine whether the quadrilateral is a
parallelogram. Justify your answer.
Answer: Each pair of opposite sides has the same
measure. Therefore, they are congruent.
If both pairs of opposite sides of a
quadrilateral are congruent, the quadrilateral
is a parallelogram.
Which method would prove the
quadrilateral is a parallelogram?
A. Both pairs of opp. sides ||.
B. Both pairs of opp. sides .
C. Both pairs of opp. s .
D. One pair of opp. sides both
|| and .
Use Parallelograms to Prove
Relationships
MECHANICS Scissor lifts, like
the platform lift shown, are
commonly applied to tools
intended to lift heavy items. In the
diagram, A  C and B  D.
Explain why the consecutive
angles will always be
supplementary, regardless of the
height of the platform.
Use Parallelograms to Prove
Relationships
Answer: Since both pairs of opposite angles of
quadrilateral ABCD are congruent, ABCD is
a parallelogram by Theorem 6.10. Theorem
6.5 states that consecutive angles of
parallelograms are supplementary.
Therefore, mA + mB = 180 and
mC + mD = 180. By substitution,
mA + mD = 180 and mC + mB = 180.
The diagram shows a car jack used to raise a car
from the ground. In the diagram, AD  BC and
AB  DC. Based on this information, which
statement will be true, regardless of the height of
the car jack.
A. A  B
B. A  C
C. AB  BC
D. mA + mC = 180
Use Parallelograms and Algebra to Find Values
Find x and y so that the quadrilateral is a
parallelogram.
Opposite sides of a parallelogram are congruent.
Use Parallelograms and Algebra to Find Values
AB = DC
Substitution
Distributive Property
Subtract 3x from each side.
Add 1 to each side.
Use Parallelograms and Algebra to Find Values
Substitution
Distributive Property
Subtract 3y from each side.
Add 2 to each side.
Answer: So, when x = 7 and y = 5, quadrilateral
ABCD is a parallelogram.
Find m so that the quadrilateral is a parallelogram.
A. m = 2
B. m = 3
C. m = 6
D. m = 8
Parallelograms and Coordinate Geometry
COORDINATE GEOMETRY
Quadrilateral QRST has vertices
Q(–1, 3), R(3, 1), S(2, –3), and
T(–2, –1). Determine whether the
quadrilateral is a parallelogram.
Justify your answer by using the
Slope Formula.
If the opposite sides of a quadrilateral are parallel,
then it is a parallelogram.
Parallelograms and Coordinate Geometry
Answer: Since opposite sides have the same slope,
QR║ST and RS║TQ. Therefore, QRST is a
parallelogram by definition.
Graph quadrilateral EFGH with vertices E(–2, 2),
F(2, 0), G(1, –5), and H(–3, –2). Determine whether
the quadrilateral is a parallelogram.
A. yes
B. no
Parallelograms and Coordinate Proofs
Write a coordinate proof for the following
statement.
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Step 1
Position quadrilateral ABCD on the coordinate
plane such that AB  DC and AD  BC.
● Begin by placing the vertex A at the origin.
● Let AB have a length of a units. Then B has
coordinates (a, 0).
Parallelograms and Coordinate Proofs
● So that the distance from D to C is also a units, let
the x-coordinate of D be b and of C be b + a.
● Since AD  BC, position the endpoints of DC so that
they have the same y-coordinate, c.
Parallelograms and Coordinate Proofs
Step 2
Use your figure to write a proof.
Given:
quadrilateral ABCD, AB  DC, AD  BC
Prove:
ABCD is a parallelogram.
Coordinate Proof:
By definition, a quadrilateral is a parallelogram if
opposite sides are parallel.
Use the Slope Formula.
Parallelograms and Coordinate Proofs
The slope of AB is 0.
The slope of CD is 0.
Since AB and CD have the same slope and AD and BC
have the same slope, AB║CD and AD║BC.
Answer: So, quadrilateral ABCD is a parallelogram
because opposite sides are parallel.
Which of the following can be used to prove the
statement below?
If a quadrilateral is a
parallelogram, then one pair of
opposite sides is both parallel and
congruent.
A. AB = a units and DC = a units;
slope of AB = 0 and slope of
DC = 0
B. AD = c units and BC = c units;
slope of
and slope of