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5.3 – Tests for Parallelograms

Key Terms

  • Opposite Sides – non-consecutive sides.
    • Not side-by-side
    • The opposite sides of a parallelogram are congruent because the opposite sides are parallel

 

Review

GeoB 5.3 Parallelograms

 

  • Determining if a Polygon is a Parallelogram
    • When you are asked, “Is this a parallelogram?”
    • You should choose “possibly” if:
      • You don’t have enough information to prove it is a parallelogram.
      • None of the information you have proves that it is not a parallelogram.
    • Do NOT choose “True” unless you are 100% certain that it’s ALWAYS true.
    • Choose “False” if you are not 100% certain.
      • If something is “possible,” choose “False” because possible truths are not ALWAYS true.
      • If something is “True” sometimes, choose “False” because it is NOT true ALL the time!

 


Notes

  • Sides
    • If BOTH pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram
    • If BOTH pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram
    • If ONE pair of opposite sides is BOTH parallel AND congruent, then it is a parallelogram
    • Test it out for yourself!  Click here!

GeoB 5.3 Parallelogram Tests


  • Angles
    • If BOTH pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram
    • If ALL pairs of consecutive angles are supplementary (add up to 180°), then it is a parallelogram

GeoB 5.3 Parallelogram by Angle


  • Diagonals
    • If BOTH diagonals bisect each other, then it is a parallelogram

GeoB 5.3 Parallelogram by Diagonals


  • Special Cases
    • If you have one pair of congruent sides AND one pair of congruent opposite angles, you can prove the quadrilateral is a parallelogram.
    • If you have one pair of congruent sides AND the SAME pair of parallel sides, you can prove the quadrilateral is a parallelogram.
    • If you have one pair of parallel sides AND one pair of alternate interior angles, you can prove the quadrilateral is a parallelogram (based on the proof in the video).

  • In the “Test” tables (below), only ONE of the “if” statements needs to be true to prove that a quadrilateral is a parallelogram

GeoB 5.3 Parallelograms If Then 1

 

GeoB 5.3 Parallelograms If Then 2


Example: Consecutive Angles of a Parallelogram are Supplementary

GeoB 5.3 Ex Consecutive

  • If RAIL is a parallelogram, what is the value of x?
    • Since consecutive angles are supplementary, 37 + 7x + 3 = 180
    • Add constant terms: 40 + 7x = 180
    • Subtract 40 from both sides: 7x = 140
    • Divide both sides by 7: x = 20

 


Example: Opposite Sides of a Parallelogram are Congruent

GeoB 5.3 Ex Congruent Sides

  • If UVWX is a parallelogram, what is the value of y?
    • Since congruent sides are equal: 13=\frac{1}{3}y
    • Multiply by 3 on both sides: 39 = y

 


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