splash screen. lesson menu five-minute check (over lesson 6–2) ccss then/now theorems: conditions...

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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B.

C.

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Over Lesson 6–2

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B.

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Over Lesson 6–2

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.

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.

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, 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.

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. mA + mC = 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

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. 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 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.

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, AB║CD and AD║BC.

The slope of CD is 0.

The slope of AB is 0.

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