.. index:: proof, triangles

.. _I.8:
.. _triangles with equal sides have equal angles:

triangles with equal sides have equal angles
============================================

  I.8

  If two triangles have the two sides equal to two sides respectively, and have
  also the base equal to the base, they will also have the angles equal which
  are contained by the equal straight lines. 

  -- Euclid


.. image:: elem.1.prop.8.png
   :align: right
   :width: 300px

Let **ABC**, **DEF** be two triangles having the two sides **AB**, **AC** equal
to the two sides **DE**, **DF** respectively, namely **AB** to **DE**, and
**AC** to **DF**; and let them have the base **BC** equal to the base **EF**;

.. image:: elem.1.prop.8.b.png
   :align: right
   :width: 300px

I say that the angle **BAC** is also equal to the angle **EDF**.

For, if the triangle **ABC** be applied to the triangle **DEF**, and if the
point **B** be placed on the point **E** and the straight line **BC** on
**EF**, 

- the point **C** will also coincide with **F**, because **BC** is equal to
  **EF**.

Then, **BC** coinciding with **EF**, 

- **BA**, **AC** [1]_ will also coincide with **ED**, **DF**;

for, if the base **BC** coincides with the base **EF**, and the sides **BA**,
**AC** do not coincide with **ED**, **DF** but fall beside them [2]_ as **EG**,
**GF**, 

- then, given two straight lines constructed on a straight line (from its
  extremities) and meeting in a point, there will have been constructed on the
  same straight line (from its extremities), and on the same side of it, two
  other straight lines meeting in another point and equal to the former two
  respectively, namely each to that which has the same extremity with it. But
  they cannot be so constructed. [I.7]

Therefore it is not possible that, if the base **BC** be applied to the base
**EF**, the sides **BA**, **AC** should not coincide with **ED**, **DF**; 

- they will therefore coincide,

so that the angle **BAC** will also coincide with the angle **EDF**, and will
be equal to it.

If therefore etc.

- Q. E. D.

dependencies
------------

[I.7]: /elem.1.7 "Book 1 - Proposition 7"

footnotes
---------


.. [1] BA, AC.

    The text has here <quote>**BA**, **CA**.</quote>

.. [2] fall beside them
   
   The Greek has the future, <foreign lang="greek">παραλλάξουσι.
   παραλλάττω</foreign> means <quote>to pass by without touching,</quote>
   <quote>to miss</quote> or <quote>to deviate.</quote>