.. index:: proof, parallelograms

.. _I.34:
.. _parallelogram angles:

parallelogram angles
====================

  I.34
  
  In parallelogrammic areas [1]_ the opposite sides and angles are equal to one
  another, and the diameter bisects the areas.

  -- Euclid

.. image:: elem.1.prop.34.png
   :align: right
   :width: 300px

Let `ACDB` be a parallelogrammic area, and `BC` its diameter; I say that the
opposite sides and angles of the parallelogram `ACDB` are equal to one another,
and the diameter `BC` bisects it. 

For, since `AB` is parallel to `CD`, and the straight line `BC` has fallen
upon them, 

- the alternate angles `ABC`, `BCD` are equal to one another. [I.29]

Again, since `AC` is parallel to `BD`, and `BC` has fallen upon them, 

- the alternate angles `ACB`, `CBD` are equal to one another. [I.29]

Therefore `ABC`, `DCB`[2]_ are two triangles having the two angles `ABC`, `BCA`
equal to the two angles `DCB`, `CBD` respectively, and one side equal to one
side, namely that adjoining the equal angles and common to both of them, `BC`; 

- therefore they will also have the remaining sides equal to the remaining
  sides respectively, and the remaining angle to the remaining angle; [I.26]
  therefore the side `AB` is equal to `CD`, and `AC` to `BD`,

and further the angle `BAC` is equal to the angle `CDB`.

And, since the angle `ABC` is equal to the angle `BCD`, 

- and the angle `CBD` to the angle `ACB`, the whole angle `ABD` is equal to the
  whole angle `ACD`. [I.c.n.2]

And the angle `BAC` was also proved equal to the angle `CDB`.

Therefore in parallelogrammic areas the opposite sides and angles are equal to
one another.

I say, next, that the diameter also bisects the areas.

For, since `AB` is equal to `CD`, and `BC` is common, the two sides `AB`, `BC`
are equal to the two sides `DC`, `CB` respectively; 

- and the angle `ABC` is equal to the angle `BCD`; therefore the base `AC` is
  also equal to `DB`, and the triangle `ABC` is equal to the triangle `DCB`.
  [I.4]

Therefore the diameter `BC` bisects the parallelogram `ACDB`.

- Q. E. D.

references
----------

[I.4]: /elem.1.4 "Book 1 - Proposition 4"
[I.26]: /elem.1.26 "Book 1 - Proposition 26"
[I.29]: /elem.1.29 "Book 1 - Proposition 29"
[I.c.n.2]: /elem.1.c.n.2 "Book 1 - Common Notion 2"

footnotes
---------

.. [1] parallelogrammic areas
    It is to be observed that, when parallelograms have to be mentioned for the
   first time, Euclid calls them <quote>parallelogrammic areas</quote> or, more
   exactly, <quote>parallelogram</quote> areas (<foreign
   lang="greek">παραλληλόγραμμα χωρία</foreign>). The meaning is simply areas
   bounded by parallel straight lines with the further limitation placed upon
   the term by Euclid that only <em>four-sided</em> figures are so called,
   although of course there are certain regular polygons which have opposite
   sides parallel, and which therefore might be said to be areas bounded by
   parallel straight lines. We gather from Proclus (<xref n="Proc. p. 393"
   from="ROOT" to="DITTO">p. 393</xref>) that the word
   <quote>parallelogram</quote> was first introduced by Euclid, that its use
   was suggested by <a href="/elem.1.33">I. 33</a>, and that the formation of
   the word <foreign lang="greek">παραλληλόγραμμος</foreign> (parallel-lined)
   was analogous to that of <foreign lang="greek">εὐθύγραμμος</foreign>
   (straight-lined or rectilineal).

      
.. [2] DCB
   and `DC`, `CB`. The Greek has in these places <quote>`BCD`</quote> and
   <quote>`CD`, `BC`</quote> respectively. Cf. note on <a href="/elem.1.33">I.
   33</a>, lines 15, 18.