____ ? A. B. C. 5-Minute Check 1

? A. B. C. 5-Minute Check 2

? A. A B. B C. C 5-Minute Check 3

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. 5-Minute Check 4

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. Then/Now

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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. Example 1

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 . Example 1

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. Example 2

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. Example 2

The diagram shows a car jack used to raise a car from the ground 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 Example 2

Find x and y so that the quadrilateral is a parallelogram. 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. Example 3

Distributive Property Use Parallelograms and Algebra to Find Values AB = DC Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Example 3

Distributive Property 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. Example 3

Find m so that the quadrilateral is a parallelogram. B. m = 3 C. m = 6 D. m = 8 Example 3

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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. Example 4

Parallelograms and Coordinate Geometry Answer: Since opposite sides have the same slope, QR║ST and RS║TQ. Therefore, QRST is a parallelogram by definition. Example 4

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 Example 4