4.2 Proving that a Quadrilateral is a Parallelogram
By the definition of a parallelogram, we know that if both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. There are other ways to prove that a quadrilateral is a parallelogram and here we establish theorems to help us do this.
- Theorem 4.5:
If both pairs of opposite sides of a quadrilateral are congruent, then
the quadrilateral is a parallelogram.
Proof (outline): By introducing a diagonal and proving that the two triangles formed are congruent, we can get pairs of alternate interior angles that are congruent. This leads to opposite sides of the quadrilateral being parallel which makes the quadrilateral a parallelogram.
- Theorem 4.6: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.
Proof (outline): Again, by introducing a diagonal and proving congruent triangles, we can show that both pairs of opposite sides of the quadrilateral are congruent; then we have a parallelogram by Theorem 4.5.
- Theorem 4.7: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Given: $\angle A\cong
\angle C$ and $\angle B\cong \angle D$ Prove: $ABCD$ is a parallelogram. |
Proof: By Theorem 3.14 we know that the sum of the measures of the angles of a quadrilateral is 360. Therefore, $m\angle A+m\angle B+m\angle C+m\angle D=360$.
Since we are given that both pairs of opposite angles of quadrilateral $ABCD$ are congruent, $m\angle A=m\angle C$ and $m\angle B=m\angle D$. Thus $2m\angle A+2m\angle D=360$; then $m\angle A+m\angle D=180$. We can show in the same way that $m\angle C+m\angle D=180$. Since same-side interior angles are hence supplementary, $\overline{AB}\parallel\overline{CD}$ and $\overline{AD}\parallel\overline{BC}$. This makes $ABCD$ a parallelogram, by definition.
- Theorem 4.8:
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
The proof of Theorem 4.8 is similar to the proofs of Theorems 4.5 and 4.6 and will be left as an exercise.
Often we say that parallel lines are "everywhere equidistant." The following theorem will establish this idea for us.
- Theorem 4.9:
If $\ell$ and $m$ are two parallel lines and points $A$ and $B$ are on
$\ell$, then the distance from $A$ to line $m$ is equal to the distance
from $B$ to line $m$.
Given: $\ell\parallel
m$, $\overline{AC}\perp m$, and $\overline{BD}\perp m$ Prove: $\overline{AC}\cong \overline{BD}$ |
Proof: In Chapter 3, we established that if two lines were perpendicular to the same line, then they were parallel to each other (Theorem 3.3). This means that $\overline{AC}\parallel\overline{BD}$. We are given that $\ell \parallel m$; therefore $\overline{AB}\parallel\overline{CD}$. Since both pairs of opposite sides of the quadrilateral $ABCD$ are parallel, by definition the quadrilateral is a parallelogram. Opposite sides of a parallelogram are congruent (Theorem 4.1). Hence $\overline{AC}\cong\overline{BD}$.
Example: Given that planes $m$ and $n$ are parallel and $\overline{AC}\parallel\overline{BD}$, prove that $ABDC$ is a parallelogram.
Solution:
Since $\overline{AC}\parallel\overline{BD}$, there is a plane
containing the points $A$, $B$, $C$, and $D$. This plane
intersects the two parallel planes $m$ and $n$. Therefore the
lines of intersection will be parallel. Hence
$\overline{AB}\parallel\overline{CD}$. Since both pairs of
opposite sides of quadrilateral $ABDC$ are parallel, the quadrilateral
is a parallelogram. |
As we did at the end of the last section, we collect here the properties that a quadrilateral might have that would allow you to claim that it is a parallelogram.
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What is really critical is that you understand the distinction between the list above and the list in Section 4.1. They look very much alike, but there is a big difference. If you write as a reason in a proof, "Opposite sides of a parallelogram are congruent," you must know already (from a previous step) that the quadrilateral is a parallelogram. If you are justifying that a quadrilateral is a parallelogram (what the current section is all about), then you write something along the lines of, "If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram." When you write, "Property of a parallelogram," you are writing about something already known to be a parallelogram — not something you are trying to prove is a parallelogram.