Opposite Sides and Angles of Parallelogram are Equal

Theorem

The opposite sides and angles of a parallelogram are equal to one another, and either of its diameters bisects its area.

In the words of Euclid:

In parallelogrammic areas the opposite sides and angles are equal to one another, the diameter bisects the areas.

(The Elements: Book $\text{I}$: Proposition $34$)

Proof

Let $ACDB$ be a parallelogram, and let $BC$ be a diameter.

By definition of parallelogram, $AB \parallel CD$, and $BC$ intersects both.

So by Parallelism implies Equal Alternate Angles:

$\angle ABC = \angle BCD$

Similarly, by definition of parallelogram, $AC \parallel BD$, and $BC$ intersects both.

So by Parallelism implies Equal Alternate Angles:

$\angle ACB = \angle CBD$

So $\triangle ABC$ and $\triangle DCB$ have two angles equal, and the side $BC$ in common.

So by Triangle Angle-Side-Angle Congruence:

$\triangle ABC = \triangle DCB$

So $AC = BD$ and $AB = CD$.

Also, we have that $\angle BAC = \angle BDC$.

So we have $\angle ACB = \angle CBD$ and $\angle ABC = \angle BCD$.

So by Common Notion $2$:

$\angle ACB + \angle BCD = \angle ABC + \angle CBD$

So $\angle ACD = \angle ABD$.

So we have shown that opposite sides and angles are equal to each other.

Now note that $AB = CD$, and $BC$ is common, and $\angle ABC = \angle BCD$.

So by Triangle Side-Angle-Side Congruence:

$\triangle ABC = \triangle BCD$

So $BC$ bisects the parallelogram.

Similarly, $AD$ also bisects the parallelogram.

$\blacksquare$

Historical Note

This proof is Proposition $34$ of Book $\text{I}$ of Euclid's The Elements.
The use of Triangle Side-Angle-Side Congruence in this proof seems to be superfluous as the triangles were already shown to be equal using Triangle Angle-Side-Angle Congruence. However, Euclid included the step in his proof, so the line is included here.

Note that in at least some translations of The Elements, the Triangle Side-Angle-Side Congruence proposition includes the extra conclusion that the two triangles themselves are equal whereas the others do not explicitly state this, but since Triangle Side-Angle-Side Congruence is used to prove the other congruence theorems, this conclusion would seem to be follow trivially in those cases.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions
• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.22$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): parallelogram