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

• 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

• 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

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

• 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

Example: Consecutive Angles of a Parallelogram are Supplementary

• 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

• 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