# 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 Equality:

- $\triangle ABC = \triangle DCB$

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 Equality:

- $\triangle ABC = \triangle BCD$

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 Equality in this proof seems to be superfluous as the triangles were already shown to be equal using Triangle Angle-Side-Angle Equality. 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 Equality 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 Equality 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**