You have probably seen that ambiguous drawing where a duck is also a rabbit, depending on how you look at it. Sometimes in geometry, something you think you know, like a parallelogram, may really be something else, like only a quadrilateral. Here are six ways to mathematically prove a given quadrilateral is a parallelogram.
Quadrilaterals are four-sided, closed polygons. Their sides can be of any length, their angles either congruent or not. The term "quadrilateral" is the loosest possible classification of a four-sided polygon.
[insert drawing of irregular quadrilateral BEAR]
If you were to go around this shape in a clockwise direction starting at ∠B, you would next get to ∠E. Those two angles are consecutive. So are all these pairs:
Consecutive angles have endpoints of the same side of the polygon.
Supplementary angles are two angles adding to 180°. In a parallelogram, any two consecutive angles are supplementary, no matter which pair you pick.
Parallelograms are special types of quadrilaterals with opposite sides parallel. Parallelograms have these identifying properties:
Parallelograms get their names from having two pairs of parallel opposite sides.
Another interesting, and useful, feature of parallelograms tells us that any angle of the parallelogram is supplementary to the consecutive angles on either side of it.
We can use these features and properties to establish six ways of proving a quadrilateral is a parallelogram.
Take a look at this quadrilateral:
[insert drawing of quadrilateral where opposite sides are very slightly not parallel and of equal length, so it is really hard to see if it is a parallelogram]
Is this quadrilateral a parallelogram? Can you be certain? Only by mathematically proving that the shape has the identifying properties of a parallelogram can you be sure. You can prove this with either a two-column proof or a paragraph proof.
Here are the six ways to prove a quadrilateral is a parallelogram:
We can use one of these ways in a two-column proof. Bear in mind that, to challenge you, most problems involving parallelograms and proofs will not give you all the information about the presented shape. Here, for example, you are given a quadrilateral and told that its opposite sides are congruent.
[insert drawing of quadrilateral GOAT with sides GO ≅ TA and TG ≅ OA]
Theorem: If a transversal cuts across two lines and the alternate interior angles are congruent, then the lines are parallel
▱ GOAT Definition of a parallelogram: A quadrilateral
with two pairs of opposite sides parallel
The two-column proof proved the quadrilateral is a parallelogram by proving opposite sides were parallel.
You can also use the paragraph proof form for any of the six ways. Paragraph proofs are harder to write because you may skip a step or leave out an explanation for one of your statements. You may wish to work very slowly to avoid problems.
Always start by making a drawing so you know exactly what you are saying about the quadrilateral as you prove it is a parallelogram.
Here is a proof still using opposite sides parallel, but with a different set of given facts. You are given a quadrilateral with diagonals that are identified as bisecting each other.
[insert drawing of quadrilateral FISH with diagonals HI and FS, but make quadrilateral clearly NOT a parallelogram; show congruency marks on the two diagonals showing they are bisected]
[Reference drawing inserted for visual clarity ONLY; it comes from https://www.homeschoolmath.net/teaching/two-column-proof.php]
Given a quadrilateral FISH with bisecting diagonals FS and HI, we can also say that the angles created by the intersecting diagonals are congruent. They are congruent because they are vertical angles (opposite angles sharing a vertex point).
Notice that we have two sides and an angle of both triangles inside the quadrilateral. So, we can use the Side-Angle-Side Congruence Theorem to declare the two triangles congruent.
Corresponding parts of congruent triangles are congruent (CPCTC), so ∠IFS and ∠ HSF are congruent. Those two angles are alternate interior angles, and if they are congruent, then sides FI and SH are parallel.
You can repeat the steps to prove FH and IS parallel, which means two pairs of opposite sides are parallel. Thus, you have a parallelogram.
In both proofs, you may say that you already were given a fact that is one of the properties of parallelograms. That is true with both proofs, but in neither case was the given information mathematically proven. You began with the given and worked through the problem, but if your proof "fell apart," then the given may have been wrong.
Since neither our two-column proof or paragraph proof "fell apart," we know the givens were true, and we know the quadrilaterals are parallelograms.
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