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Conditions for Special Parallelograms

Conditions for Special Parallelograms . Geometry CP1 (Holt 6-5) K. Santos. Special Parallelograms. Special Parallelograms: Rectangle Rhombus Square. Theorem 6-5-1. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. A B

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Conditions for Special Parallelograms

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  1. Conditions for Special Parallelograms Geometry CP1 (Holt 6-5) K. Santos

  2. Special Parallelograms Special Parallelograms: Rectangle Rhombus Square

  3. Theorem 6-5-1 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. A B D C Given: ABCD is a parallelogram with right angle <A Then: ABCD is a rectangle

  4. Theorem 6-5-2 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. A B D C Given: parallelogram with Then: ABCD is a rectangle

  5. Theorem 6-5-3 If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A B D C Given: ABCD is a parallelogram with Then: ABCD is a rhombus

  6. Theorem 6-5-4 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. A B D C Given: parallelogram with Then: ABCD is a rhombus

  7. Theorem 6-5-5 If one diagonal of a parallelogram bisects a pair angles of the parallelogram, then the parallelogram is a rhombus. B C A D Given: parallelogram ABCD bisects < BAD and < BCD Then: ABCD is a rhombus

  8. Is the Parallelogram a Rhombus or Rectangle? Rhombus ---all four sides of a quadrilateral are congruent ---one pair of consecutive sides of a parallelogram are congruent ---one diagonal of a parallelogram bisects two angles of the parallelogram ---the diagonals of a parallelogram are perpendicular Rectangle ---all four angles of the quadrilateral are right angles ---the parallelogram has a right angle ---the diagonals of a parallelogram are congruent

  9. What about a Square? To prove you have a square you must prove it is both a rectangle and a rhombus.

  10. Example—Identify special parallelogram Tell whether the quadrilateral is a parallelogram, rectangle, rhombus, or square. Given al the names that apply. Diagonals—bisected, congruent and perpendicular Parallelogram, rectangle, rhombus and square

  11. Example—valid conclusion??? Determine if the conclusion is valid. FGHE is a square F G , B EBH E H Diagonals—congruent and bisected possible rectangle/rhombus---square Opposite sides congruent---parallelogram Yes—valid conclusion

  12. Example—Is it a Rhombus? Determine is ABCD is a rhombus. A(0,2), B(3,6), C(8,6) and D(5,2). Test to see if all four sides are congruent: AB=CD= AB= CD = AB = CD = AB = = 5 CD = = 5 BC=DA = BC = DA = BC = DA = BC = = 5 DA = = 5

  13. Example—using diagonals (part I) Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus or square. Give all the names that apply. P(-1,4), Q(2,6), R(4,3) and S(1,1) PR= QS = PR= QS = PR = QS = PR = QS = Determines that diagonals are congruent—so a rectangle or maybe a square

  14. Example—continued (Part II) P(-1,4), Q(2,6), R(4,3) and S(1,1) So to determine if it is a square then we need to test if the diagonals are perpendicular (slope) Slope PR slope of QS Slope PR = slope QS = Slope PR = slope QS = = 5 slope are negative reciprocals—perpendicular rhombus Diagonals---congruent & perpendicular---rhombus So, the quadrilateral could be: rectangle, rhombus, square

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