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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 Lesson Menu

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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

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

G.CO.11 Prove theorems about parallelograms. 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. CCSS

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. 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: 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

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 Example 5

HOMEWORK Pg. 417-419 4-7,9-14, 18-23, 24,25, 52-54 Concept 3

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