6.2 Parallelograms and their properties

Parallelogram Definition and Properties

A parallelogram is a quadrilateral with two pairs of parallel sides. Its properties give you a reliable set of relationships between sides, angles, and diagonals that you can use to solve for unknowns, write proofs, and classify shapes. This section covers those core properties, the theorems behind them, and how to apply them.

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Properties of Parallelograms

In any parallelogram $ABCD$, the following properties always hold:

• Opposite sides are parallel. $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \parallel \overline{AD}$. This is the defining property.
• Opposite sides are congruent. $AB = CD$ and $BC = AD$.
• Opposite angles are congruent. $\angle A \cong \angle C$ and $\angle B \cong \angle D$.
• Consecutive angles are supplementary. Any two angles that share a side add to $180°$. For example, $\angle A + \angle B = 180°$. This follows directly from the parallel sides: $\overline{AB} \parallel \overline{CD}$ with transversal $\overline{BC}$ makes $\angle B$ and $\angle C$ co-interior (same-side interior) angles.
• Diagonals bisect each other. If diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $M$, then $AM = MC$ and $BM = MD$. The diagonals themselves don't have to be equal in length; they just cut each other in half.

One thing worth remembering: knowing any one of these properties (besides the definition itself) isn't enough to prove a shape is a parallelogram. You'll see the converse theorems in the next section, but for now, these properties are consequences of already knowing the shape is a parallelogram.

Theorems and Their Proofs

These properties aren't just observations. They're proven from the definition using triangle congruence. Both proofs below rely on the same core idea: draw a diagonal to split the parallelogram into triangles, then use alternate interior angles.

Theorem: Opposite sides of a parallelogram are congruent.

1. Start with parallelogram $ABCD$. Draw diagonal $\overline{AC}$, splitting it into two triangles: $\triangle ABC$ and $\triangle CDA$.
2. Because $\overline{AB} \parallel \overline{CD}$, the diagonal acts as a transversal. By the Alternate Interior Angles Theorem, $\angle BAC \cong \angle DCA$.
3. Similarly, because $\overline{BC} \parallel \overline{AD}$, we get $\angle BCA \cong \angle DAC$.
4. Side $\overline{AC}$ is shared by both triangles (Reflexive Property).
5. By ASA, $\triangle ABC \cong \triangle CDA$. Therefore $AB = CD$ and $BC = AD$ by CPCTC.

Notice that this same congruence also proves opposite angles are congruent: $\angle B \cong \angle D$ follows from CPCTC. Drawing the other diagonal proves $\angle A \cong \angle C$.

Theorem: Diagonals of a parallelogram bisect each other.

1. Draw both diagonals $\overline{AC}$ and $\overline{BD}$, intersecting at point $M$.
2. In $\triangle ABM$ and $\triangle CDM$: we already know $AB = CD$ (opposite sides). The alternate interior angles give us $\angle ABM \cong \angle CDM$ and $\angle BAM \cong \angle DCM$.
3. By ASA, $\triangle ABM \cong \triangle CDM$. So $AM = CM$ and $BM = DM$ by CPCTC.

Applying Parallelogram Properties

These properties let you set up equations to find unknown measurements. Here are the main strategies:

Finding side lengths: If you know one side, the opposite side is equal. If $AB = 5$ cm, then $CD = 5$ cm. If a side is given as an expression, set opposite sides equal and solve.

Finding angle measures: If you know one angle, the opposite angle is equal, and each consecutive angle is its supplement. If $\angle A = 70°$, then $\angle C = 70°$ and $\angle B = \angle D = 110°$. So knowing just one angle determines all four.

Using diagonals: If diagonal $\overline{AC}$ has total length 12 cm, then $AM = MC = 6$ cm. When you're given diagonal segments as expressions, set the two halves equal and solve.

The diagonals create two pairs of congruent triangles ($\triangle ABM \cong \triangle CDM$ and $\triangle BCM \cong \triangle DAM$), which is useful in proofs. Be careful: the four triangles formed are not all congruent to each other unless the parallelogram happens to be a rectangle.

Constructing a Parallelogram

You can construct a parallelogram given any of these sets of information:

1. Two adjacent side lengths and the included angle
2. Two adjacent side lengths and a diagonal length
3. One side and two adjacent angles

Steps for construction (given two sides and the included angle):

1. Draw side $\overline{AB}$ with the given length.
2. At point $A$, construct the given angle using a protractor or compass-and-straightedge method.
3. Along the ray from $A$, mark point $D$ at the second given side length.
4. From $B$, construct a line parallel to $\overline{AD}$. You can do this by copying $\angle DAB$ at point $B$ on the opposite side of $\overline{AB}$, or by setting your compass to length $AD$ and drawing an arc from $B$.
5. From $D$, set your compass to length $AB$ and draw an arc that intersects the line from step 4. Label this intersection $C$.
6. Connect $D$ to $C$ to complete the parallelogram.
7. Verify your result: check that opposite sides are congruent and diagonals bisect each other.