Download 6-3 Tests for Parallelograms 1-6

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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
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Over Lesson 6–2
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Over Lesson 6–2
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A.
B.
C.
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Over Lesson 6–2
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A.
B.
C.
Over Lesson 6–2
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A. ∠A
B. ∠B
C. ∠C
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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.
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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.
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Identify Parallelograms
A quadrilateral is a parallelogram if:
1. Both pairs of opposite sides are congruent
2. Both pairs of opposite angles are congruent
3. The diagonals bisect each other
4. One pair of opposite sides are both congruent
and parallel.
5. Both pairs of opposite sides are parallel
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 ≅.
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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.
Answer If two parallel lines
are cut by a transversal then
consecutive interior angles
are supplementary.
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
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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.
AB = DC
y=5
Find m so that the quadrilateral is a parallelogram.
A. m = 2
B. m = 3
C. m = 6
D. m = 8
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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.
∆
∆
∆
∆
=
=
=
=
∆
∆
∆
∆
4
−2
4
−2
=
=
=
=
1
4
1
4
1
1
= − = − = 4 = 4 2
2
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.
Answer: Since opposite sides have the same slope,
QR║ST and RS║TQ. Therefore, QRST is a
parallelogram by definition.
= 4
= 4 = −
1
1
= − 2
2
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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.
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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 2 Write the proof
∆
−
0
=
=
=0
∆
+ −0 +
∆ 0 − 0 0
= =
= =0
∆ − 0 ∆ − 0 = =
=
∆ − 0 ∆
−0
= =
=
∆ ( + ) − = Since both pairs of opposite sides are parallel, the
quadrilateral is a parallelogram.”
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
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