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Splash Screen. Please read the following and consider yourself in it. I am capable of learning. I can accomplish mathematical tasks. I am ultimately responsible for my learning. An Affirmation. Unit Objectives

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  1. Splash Screen

  2. Please read the following and consider yourself in it. I am capable of learning. I can accomplish mathematical tasks. I am ultimately responsible for my learning. An Affirmation

  3. Unit Objectives G.GPE.5- SWBAT apply properties of similar triangles IOT prove slope criteria for parallel and perpendicular lines. G.GPE.5-SWBAT apply slope criteria for parallel and perpendicular lines IOT find the equation of a line parallel or perpendicular to a given line that passes through a given point. G.GPE.4 SWBAT apply distance formula, partition formula and slope criteria IOT prove geometric theorems on quadrilaterals and polygons algebraically. G.GPE.7- SWBAT apply algebraic methods on coordinates IOT to compute perimeters of polygons and areas of rectangles and triangles.

  4. Review Activity for Equations of Lines Classify the lines as parallel, perpendicular or neither with respects to each other: a. 2x + 3y =5, 3x + 2y = 7 b. x –y =4, 2x -2y = 3 c. x+2y = 5, 2x-y = 0 1. Find the equation of the line parallel to 3x-2y =5 and passing through (3, -2) 2. Find the equation of the perpendicular bisector of the line joining the points (3, -2) and ( 5, 4) 3. Find the equation of the line parallel to x =3 and passing through (5,-1) 4. Find the equation of the line perpendicular to x =3 and passing through (5,-1) Lesson Menu

  5. ? ____ A. B. C. 5-Minute Check 1

  6. ? ____ A. B. C. 5-Minute Check 1

  7. ? A. B. C. 5-Minute Check 2

  8. ? A. B. C. 5-Minute Check 2

  9. ? A. A B. B C. C 5-Minute Check 3

  10. ? A. A B. B C. C 5-Minute Check 3

  11. A.A  C and B  D B.A  B and C  D C. D. 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? 5-Minute Check 4

  12. A.A  C and B  D B.A  B and C  D C. D. 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? 5-Minute Check 4

  13. 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

  14. 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

  15. Concept 1

  16. Concept 2

  17. Identify Parallelograms Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: Example 1

  18. 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

  19. 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

  20. 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

  21. 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

  22. Use Parallelograms to Prove Relationships Answer: Example 2

  23. 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

  24. 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

  25. 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

  26. 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

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

  28. Use Parallelograms and Algebra to Find Values Substitution Distributive Property Subtract 3y from each side. Add 2 to each side. Answer: Example 3

  29. Use Parallelograms and Algebra to Find Values Substitution Distributive Property Subtract 3y from each side. Add 2 to each side. Answer: So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram. Example 3

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

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

  32. Concept 3

  33. Parallelograms and Coordinate Geometry COORDINATE GEOMETRYQuadrilateral 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

  34. Parallelograms and Coordinate Geometry Answer: Example 4

  35. Answer: Since opposite sides have the same slope, QR║ST and RS║TQ. Therefore, QRST is a parallelogram by definition. Parallelograms and Coordinate Geometry Example 4

  36. 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

  37. 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

  38. Step 1 Position 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). 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. Example 5

  39. ● Since AD BC, position the endpoints of DC so that they have the same y-coordinate, c. 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. Example 5

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

  41. The slope of AB is 0. The slope of CD is 0. Since AB and CD have the same slope and AD and BC have the same slope, AB║CD and AD║BC. Parallelograms and Coordinate Proofs Answer: Example 5

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

  43. 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 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. Example 5

  44. 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 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. Example 5

  45. End of the Lesson

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