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Splash Screen. Lesson Menu Five-Minute Check (over Lesson 6–2) CCSS Then/Now Theorems: Conditions for Parallelograms Proof: Theorem 6.9 Example 1: Identify.

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Presentation on theme: "Splash Screen. Lesson Menu Five-Minute Check (over Lesson 6–2) CCSS Then/Now Theorems: Conditions for Parallelograms Proof: Theorem 6.9 Example 1: Identify."— Presentation transcript:

1 Splash Screen

2 Lesson Menu 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

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

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

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

6 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.

7 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.

8 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.

9 Concept 1

10 Concept 2

11 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.

12 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?

13 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.

14 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.

15 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

16 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.

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

18 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.

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

20 Concept 3

21 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.

22 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.

23 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.

24 Example 5 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. ●Begin by placing the vertex A at the origin. Step 1Position quadrilateral ABCD on the coordinate plane such that AB  DC and AD  BC. ●Let AB have a length of a units. Then B has coordinates (a, 0).

25 Example 5 Parallelograms and Coordinate Proofs ●So that the distance from D to C is also a units, let the x-coordinate of D be b and of C be b + a. ●Since AD  BC, position the endpoints of DC so that they have the same y-coordinate, c.

26 Example 5 Parallelograms and Coordinate Proofs Step 2Use your figure to write a proof. Given:quadrilateral ABCD, AB  DC, AD  BC Prove:ABCD is a parallelogram. Coordinate Proof: By definition, a quadrilateral is a parallelogram if opposite sides are parallel. Use the Slope Formula.

27 Example 5 Parallelograms and Coordinate Proofs Answer:So, quadrilateral ABCD is a parallelogram because opposite sides are parallel. Since AB and CD have the same slope and AD and BC have the same slope, AB║CD and AD║BC. The slope of CD is 0. The slope of AB is 0.

28 Example 5 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

29 End of the Lesson


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