Key takeaway: A parallelogram is a four-sided figure whose opposite sides are parallel in both pairs, which forces equal opposite sides, equal opposite angles and diagonals that bisect one another.
Grade 10 – GeometryG-CO.C.11Geometry and Measurement
Definition of Parallelogram
Picture a rectangle pushed over sideways: that leaning shape is a parallelogram. Its opposite sides keep running in the same direction, so each pair is parallel.
Several properties follow. Opposite sides have equal length, opposite angles have equal measure, and angles that sit next to each other add up to \(180^{\circ}\). The two diagonals cut each other in half at their crossing point, but they are usually unequal. Rectangles, rhombuses and squares are all special parallelograms.
Formula and notation
Name it by its vertices in order, \(\square ABCD\), with \(\overline{AB} \parallel \overline{DC}\) and \(\overline{AD} \parallel \overline{BC}\). With diagonals meeting at \(E\):
\[ AB = DC,\quad AD = BC,\quad AE = EC,\quad BE = ED \]
Its area is base times height: \(A = b\,h\).
Key facts
- Consecutive angles are supplementary, while opposite angles are equal.
- The diagonals bisect each other, so each one is cut at its midpoint.
- One pair of sides that is both equal and parallel is enough to prove it.
- The height is measured at a right angle to the base, not along the slanted side.
Parallelogram example
In parallelogram \(ABCD\) the diagonals meet at \(E\). If \(AE = 4x – 3\), \(EC = 2x + 9\) and \(m\angle A = 74^{\circ}\), find the diagonal \(AC\) and the measures of \(\angle B\) and \(\angle C\).
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The crossing point is the middle of each diagonal, so \(AE = EC\): \(4x – 3 = 2x + 9\), which leaves \(2x = 12\) and \(x = 6\).
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Then \(AE = 21\), \(EC = 21\) and \(AC = 42\).
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Consecutive angles add to \(180^{\circ}\): \(m\angle B = 180 – 74 = 106^{\circ}\). Opposite angles are equal: \(m\angle C = 74^{\circ}\).
\(AC = \) 42, \(m\angle B = \) \(106^{\circ}\) and \(m\angle C = \) \(74^{\circ}\).
Common mistake
Watch out
Omar says the diagonals of a parallelogram must be equal in length, because they cross at their midpoints.
Why it is wrong
Crossing at midpoints only says each diagonal is halved. Equal diagonals happen in rectangles. In a parallelogram with angles of \(74^{\circ}\) and \(106^{\circ}\), one diagonal is longer than the other.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement