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 1
Over Lesson 6 2??? Over Lesson 6 2? A. B. C. 2
Over Lesson 6 2? A. B. C. Over Lesson 6 2? A. A B. B C. C 3
Over Lesson 6 2 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. 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. 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. 5
Identify Parallelograms A quadrilateral is a parallelogram if: 1. Both pairs of opposite sides are congruent 2. Both pairs of opposite angles are congruent 3. The diagonals bisect each other 4. One pair of opposite sides are both congruent and parallel. 5. Both pairs of opposite sides are parallel 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. 6
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. Answer If two parallel lines are cut by a transversal then consecutive interior angles are supplementary. Use Parallelograms to Prove Relationships 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 7
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. AB = DC y = 5 Find m so that the quadrilateral is a parallelogram. A. m = 2 B. m = 3 C. m = 6 D. m = 8 8
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. = 4 1 =4 = 4 1 =4 = 2 4 = 2 4 = 1 2 = 1 2 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. Answer: Since opposite sides have the same slope, QR ST and RS TQ. Therefore, QRST is a parallelogram by definition. =4 =4 = 1 2 = 1 2 9
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 Parallelograms and Coordinate Proofs Write a coordinate proof for the following statement. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Step 1 Position quadrilateral ABCD on the coordinate plane such that AB DC and AD BC. 10
Parallelograms and Coordinate Proofs Write a coordinate proof for the following statement. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Step 2 Write the proof = + 0 = 0 + =0 =0 0 0 =0 =0 = 0 0 = = 0 ( + ) = Since both pairs of opposite sides are parallel, the quadrilateral is a parallelogram. 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 11
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