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.

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.