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Understand Parallelograms: Conditions, Proofs, and Properties

Learn about conditions establishing a parallelogram, solve problems using properties, and discover rectangle properties and its relation to a parallelogram. Practice with examples and proofs in geometry concepts.

jeffreyavery
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Understand Parallelograms: Conditions, Proofs, and Properties

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  1. Lesson: 6.3 Tests for Parallelograms Objectives: • To Identify the 5 CONDITIONS that GUARANTEE that a QUADRILATERAL is a PARALLELOGRAM • To Use the 5 CONDITIONS to SOLVE Problems

  2. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN:

  3. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN: 1. OPPOSITE SIDES are PARALLEL

  4. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN: 1. OPPOSITE SIDES are PARALLEL 2. OPPOSITE SIDES are CONGRUENT.

  5. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN: 1. OPPOSITE SIDES are PARALLEL 2. OPPOSITE SIDES are CONGRUENT. 3. OPPOSITE ANGLES are CONGRUENT.

  6. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN: 1. OPPOSITE SIDES are PARALLEL 2. OPPOSITE SIDES are CONGRUENT. 3. OPPOSITE ANGLES are CONGRUENT. 4. CONSECUTIVE ANGLES are SUPPLEMENTARY.

  7. GEOMETRY 6.3 IF a Quadrilateral is a Parallelogram THEN: 1. OPPOSITE SIDES are PARALLEL 2. OPPOSITE SIDES are CONGRUENT. 3. OPPOSITE ANGLES are CONGRUENT. 4. CONSECUTIVE ANGLES are SUPPLEMENTARY. 5. DIAGONALS Bisect each other.

  8. GEOMETRY 6.3 Which, if any, of the Properties of a Parallelogram PROVE that a Quadrilateral IS a Parallelogram?

  9. GEOMETRY 6.3 IF a QUADRILATERAL has OPPOSITE SIDES that are PARALLEL Is it a PARALLELOGRAM?

  10. GEOMETRY 6.3 IF a QUADRILATERAL has OPPOSITE SIDES that are PARALLEL Is it a PARALLELOGRAM? YES – the DEFINITION of a PARALLELOGRAM is a Quadrilateral for which OPPOSITE SIDES are Parallel!

  11. GEOMETRY 6.3 IF a QUADRILATERAL has ONE PAIR of OPPOSITE SIDES that are CONGRUENT, Is it a PARALLELOGRAM?

  12. GEOMETRY 6.3 IF a QUADRILATERAL has ONE PAIR of OPPOSITE SIDES that are CONGRUENT, Is it a PARALLELOGRAM? Can you DRAW a COUNTEREXAMPLE?

  13. GEOMETRY 6.3 IF a QUADRILATERAL has BOTH PAIR of OPPOSITE SIDES that are CONGRUENT, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE?

  14. GEOMETRY 6.3 IF a QUADRILATERAL has BOTH PAIR of OPPOSITE SIDES that are CONGRUENT, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE? Can you PROVE it?

  15. GEOMETRY 6.3 IF a QUADRILATERAL has ONE PAIR of OPPOSITE ANGLES that are CONGRUENT, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE? Can you PROVE it?

  16. GEOMETRY 6.3 IF a QUADRILATERAL has BOTH PAIRs of OPPOSITE ANGLES that are CONGRUENT, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE? Can you PROVE it?

  17. GEOMETRY 6.3 GEOMETRY 6.3 Given: Angles T and R are Congruent Angles Q and S are Congruent Prove: QRST is a Parallelogram

  18. GEOMETRY 6.3 IF a QUADRILATERAL has DIAGONALS that Bisect each other, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE? Can you PROVE it?

  19. GEOMETRY 6.3

  20. GEOMETRY 6.3 IF a QUADRILATERAL has ONE PAIR of OPPOSITE SIDES that is BOTH PARALLEL and CONGRUENT, Is it a PARALLELOGRAM? Is there a COUNTEREXAMPLE? Can you PROVE it?

  21. GEOMETRY 6.3

  22. GEOMETRY 6.3

  23. GEOMETRY 6.3

  24. GEOMETRY 6.3

  25. GEOMETRY 6.3

  26. GEOMETRY 6.3

  27. GEOMETRY 6.3

  28. GEOMETRY 6.3

  29. GEOMETRY 6.3

  30. COORDINATE GEOMETRYDetermine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Three Methods: 1. SLOPE formula 2. DISTANCE formula 3. MIDPOINTformula

  31. Geometry 6.3 • You should be able to: • Determine is a Quadrilateral is a PARALLEOGRAM • Determine if a CONDITION defines a PARALLELOGRAM

  32. Lesson: 6.4 Rectangles Objectives: • To Identify the PROPERTIES of RECTANGLES • To Use the Rectangle Properties to SOLVE Problems • To Identify the PROPERTIES of SQUARES and RHOMBI • To use the Squares and Rhombi Properties to SOLVE Problems

  33. GEOMETRY 6.4 A RECTANGLE is:

  34. GEOMETRY 6.4 GEOMETRY 6.4 A RECTANGLE is: A QUADRILATERAL

  35. GEOMETRY 6.4 GEOMETRY 6.4 A RECTANGLE is: A QUADRILATERAL A PARALLELOGRAM

  36. GEOMETRY 6.4 GEOMETRY 6.4 A RECTANGLE is: A QUADRILATERAL A PARALLELOGRAM with 4 Right Angles

  37. GEOMETRY 6.4 PROPERTIES of a Rectangle: • Same as a Parallelogram

  38. GEOMETRY 6.4 PROPERTIES of a Rectangle: • Same as a Parallelogram • Opposite Sides are Parallel • Opposite Sides are Congruent • Opposite Angles are Congruent • Consecutive Sides are Supplementary • Diagonals BISECT each other.

  39. GEOMETRY 6.4 PROPERTIES of a Rectangle: • Same as a Parallelogram • Opposite Sides are Parallel • Opposite Sides are Congruent • Opposite Angles are Congruent • Consecutive Sides are Supplementary • Diagonals BISECT each other. • All ANGLES are CONGRUENT

  40. GEOMETRY 6.4 PROPERTIES of a Rectangle: • Same as a Parallelogram • Opposite Sides are Parallel • Opposite Sides are Congruent • Opposite Angles are Congruent • Consecutive Sides are Supplementary • Diagonals BISECT each other. • All ANGLES are CONGRUENT • DIAGONALS are

  41. PROOF GEOMETRY 6.4 GIVEN: A B PROVE: D C • DIAGONALS are

  42. M N GEOMETRY 6.4 C P O Find X

  43. M N GEOMETRY 6.4 GEOMETRY 6.4 GEOMETRY 6.4 C P O Find X

  44. M N GEOMETRY 6.4 GEOMETRY 6.4 GEOMETRY 6.4 GEOMETRY 6.4 GEOMETRY 6.4 C P O Find X Find X

  45. GEOMETRY 6.4 GEOMETRY 6.4 TRUE or FALSE? If a QUADRILATERAL has OPPOSITE SIDES that are CONGRUENT, then it is a RECTANGLE.

  46. L K GEOMETRY 6.4 8 1 7 2 C 9 10 3 6 4 5 N M

  47. L K GEOMETRY 6.4 GEOMETRY 6.4 8 1 7 2 C 9 10 3 6 4 5 N M

  48. L K GEOMETRY 6.4 GEOMETRY 6.4 8 1 7 2 C 9 10 3 6 4 5 N M

  49. Kyle is building a barn for his horse. He measures the diagonals of the door opening to make sure that they bisect each other and they are congruent. How does he know that the measure of each corner is 90?

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