Over Lesson 6–2 5-Minute Check 1 A. B. C. ____ ?

Over Lesson 6–2 5-Minute Check 2 A. B. C. ?

Over Lesson 6–2 5-Minute Check 3 A.  A B.  B C.  C ?

Over Lesson 6–2 5-Minute Check 4 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.

Then/Now 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.

Concept 1

Concept 2

Example 1 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 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?

Example 2 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 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. 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 3 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 Use Parallelograms and Algebra to Find Values Substitution Distributive Property Add 1 to each side. Subtract 3x from each side. AB = DC

Example 3 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.

Example 3 A.m = 2 B.m = 3 C.m = 6 D.m = 8 Find m so that the quadrilateral is a parallelogram.

Concept 3

Example 4 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 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.