Five-Minute Check (over Lesson 6–2)

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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An expandable gate is made of parallelograms that have angles that change measure as the gate is adjusted. Which of the following statements about the angles in the gate is always true?A. A C and B D

B. A B and C D

C.

D.

You recognized and applied properties of parallelograms. (Lesson 6–2)

• 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.

A. A

B. B

C. C

D. D A B C D

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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 .

Which method would prove the quadrilateral is a parallelogram?

Use Parallelograms to Prove Relationships

MECHANICS Scissor lifts, like the platform lift shown below, 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, mA + mB = 180 and mC + mD = 180. By substitution, mA + mD = 180 and mC + mB = 180.

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B. B

C. C

D. D A B C D

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A. A B

B. A C

C. AB BC

D. mA + mC = 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.

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

Substitution

Distributive Property

Add 1 to each side.

Subtract 3x from each side.

AB = DC

Use Parallelograms and Algebra to Find Values

Answer: So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram.

Substitution

Distributive Property

Add 2 to each side.

Subtract 3y from each side.

A. A

B. B

C. C

D. D A B C D

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A. m = 2

B. m = 3

C. m = 6

D. m = 8

Find m so that the quadrilateral is a parallelogram.

Parallelograms and Coordinate Geometry

COORDINATE GEOMETRY Graph quadrilateral QRST with 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.

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2. B

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A. yes

B. no

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.

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.

● Begin by placing the vertex A at the origin.

Step 1 Position quadrilateral ABCD on the coordinateplane such that AB DC and AD BC.

● Let AB have a length of a units. Then B hascoordinates (a, 0).

Parallelograms and Coordinate Proofs

● So that the distance from D to C is also a units, letthe x-coordinate of D be b and of C be b + a.

● Since AD BC position the endpoints of DC so thatthey 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

Answer: So, quadrilateral ABCD is a parallelogrambecause opposite sides are parallel.

Since AB and CD have the same slope and AD and BC have the same slope, AD║BC and AB║CD.

The slope of CD is 0.

The slope of AB is 0.

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2. B

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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