.. index:: proof, triangles

.. _I.6:
.. _equal triangle angles of triangle subtend equal sides

when two angles of a triangle are equal, so are the subtended sides
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  I.6

  If in a triangle two angles be equal to one another, the sides which subtend
  the equal angles will also be equal to one another.

  -- Euclid


.. image:: elem.1.prop.6.png
   :align: right
   :width: 300px

Let **ABC** be a triangle having the angle **ABC** equal to the angle **ACB**; 

I say that the side **AB** is also equal to the side **AC**.

For, if **AB** is unequal to **AC**, one of them is greater.

Let **AB** be greater; and from **AB** the greater let **DB** be cut off equal
to **AC** the less;

let **DC** be joined. 

Then, since **DB** is equal to **AC**, and **BC** is common, 

- the two sides **DB**, **BC** are equal to the two sides **AC**, **CB**
  respectively;

and the angle **DBC** is equal to the angle **ACB**; 

- therefore the base **DC** is equal to the base **AB**, and the triangle
  **DBC** will be equal to the triangle **ACB**, the less to the greater: which
  is absurd. Therefore **AB** is not unequal to **AC**; it is therefore equal
  to it.

Therefore etc.

- Q. E. D.