Parallelogram
In short: A parallelogram is a four-cornered figure built from two pairs of parallel edges, which locks in matching opposite edges, matching opposite corners and diagonals that cut each other in half.
GeometryGeometryG-CO.C.11
What is Parallelogram?
Push on the top of a rectangular picture frame and it leans into a parallelogram: still two pairs of parallel sides, but no longer square corners. That single parallel condition is enough to guarantee a whole list of facts.
Opposite sides share a length, opposite angles share a measure, and neighboring angles sum to \(180^{\circ}\). The two diagonals slice each other in half at their crossing point. Rectangles, rhombuses and squares all belong to this family.
How to spot it
- A figure shows a slanted four-sided shape with arrowheads marking two pairs of parallel sides.
- A problem gives expressions for opposite sides or opposite angles of a quadrilateral called ABCD.
- Diagonals are drawn and the crossing point is said to split each of them in half.
- A proof asks you to show a quadrilateral qualifies, using one pair of sides that are both equal and parallel.
The method
- Match each side with the one facing it and set their lengths equal; match opposite corners the same way.
- Use \(180^{\circ}\) for any two neighboring corners to find a missing one.
- For diagonals, treat each as two equal pieces joined where the diagonals cross.
- To certify a candidate, a single criterion is enough, for instance one pair of sides that is parallel and equal.
Parallelogram: solved example
Problem
In parallelogram \(ABCD\), \(AB = 4x – 3\) and \(CD = 2x + 9\), while \(m\angle A = (5y + 10)^{\circ}\) and \(m\angle C = (7y – 30)^{\circ}\). Find \(AB\), \(m\angle A\) and \(m\angle B\).
Solution
Opposite sides are equal: \(4x – 3 = 2x + 9\), so \(x = 6\) and \(AB = 4(6) – 3 = 21\).
Opposite angles are equal: \(5y + 10 = 7y – 30\), so \(y = 20\) and \(m\angle A = 5(20) + 10 = 110\). Neighbors are supplementary, so \(m\angle B = 180 – 110 = 70\). Results: \(AB = 21\), \(110^{\circ}\) and \(70^{\circ}\).
Try it
Question 1
The diagonals of a window frame shaped like this cross at \(E\), and \(AE = 7.5\) cm. How long is diagonal \(\overline{AC}\)?
Show the answer
Each diagonal is halved at \(E\), so \(AC = 2 \times 7.5 = 15\) cm.
Question 2
One corner of a slanted tile measures \(125^{\circ}\). Find the other three corners.
Show the answer
The opposite corner is \(125^{\circ}\), and the two neighbors are \(180 – 125 = 55^{\circ}\) each.
Parallelogram vs. Rhombus
A rhombus is the special case with four equal sides, and its diagonals cross at right angles. The general version only promises equal opposite sides, so a slanted 6 by 9 example qualifies without being a rhombus. Every rhombus passes the tests above; the reverse fails.
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