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CCSS 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.

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

1. Test if the opposite sides are congruent 2. Test if the opposite angles are congruent 3. Test if diagonals bisect each other (midpoint formula) 4. Test if the slopes of opposite sides are the same (are opposite sides parallel?) 5. Determine if one pair of opposite sides are congruent AND parallel How we test if a quadrilateral is a parallelogram

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.

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.

Homework: Page #’s 1-7

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.