<PRE><B>Question 1007637</B>
.
Write a two column proof
Line segment AB IS congruent to line segment AC, angle BAD is congruent to angle CAD
Prove:line segment AD bisects line segment BC
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&lt;pre&gt;
This statement is usually formulated as the 

&lt;U&gt;Theorem&lt;/U&gt;. In an isosceles triangle, the angle bisector to the angle between congruent sides is the median. 


See the lesson &lt;A HREF=http://www.algebra.com/algebra/homework/Triangles/An-altitude-a-median-and-an-angle-bisector-in-the-isosceles-triangle.lesson&gt;An altitude a median and an angle bisector in the isosceles triangle&lt;/A&gt; in this site.


Below is the proof from this lesson.

&lt;TABLE&gt;
  &lt;TR&gt;
  &lt;TD&gt;
&lt;U&gt;Figure 3&lt;/U&gt; shows an isosceles triangle &lt;B&gt;ABC&lt;/B&gt; with sides &lt;B&gt;AB&lt;/B&gt; and &lt;B&gt;AC&lt;/B&gt; of equal length.
The segment &lt;B&gt;AD&lt;/B&gt; is the bisector of the angle &lt;B&gt;BAC&lt;/B&gt; opposite to the base &lt;B&gt;BC&lt;/B&gt;.
We need to prove that &lt;B&gt;CD&lt;/B&gt; is the median of the triangle &lt;B&gt;ABC&lt;/B&gt;.


Since &lt;B&gt;AD&lt;/B&gt; is a bisector of the angle &lt;B&gt;BAC&lt;/B&gt;, the angles &lt;B&gt;BAD&lt;/B&gt; and &lt;B&gt;CAD&lt;/B&gt; are congruent.
Thus, the triangles &lt;B&gt;ADB&lt;/B&gt; and &lt;B&gt;ADC&lt;/B&gt; have the pair of congruent sides &lt;B&gt;AB&lt;/B&gt; = &lt;B&gt;AC&lt;/B&gt;, the common &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;
side &lt;B&gt;AD&lt;/B&gt; and the congruent included angles &lt;B&gt;BAD&lt;/B&gt; and &lt;B&gt;CAD&lt;/B&gt;.
Hence, these triangles are congruent, in accordance to the &lt;B&gt;postulate P1 (SAS)&lt;/B&gt; of the triangle 
congruency (see the lesson &lt;A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson&gt; Congruence tests for triangles&lt;/A&gt; under the current topic in this site).
It implies that the segments &lt;B&gt;BD&lt;/B&gt; and &lt;B&gt;CD&lt;/B&gt; are congruent. 
This means that the bisector of the angle &lt;B&gt;ACB&lt;/B&gt; is the median. 
Thus, the statement is proved.
 &lt;/TD&gt;
  &lt;TD&gt;
{{{drawing( 200, 250,  0, 4, 0, 5, 
            line( 0.3, 0.5, 3.7, 0.5), 
            line( 0.3, 0.5, 2.0, 4.5),
            line( 2.0, 4.5, 3.7, 0.5),

            locate(0.3, 0.5, B),
            locate(3.7, 0.5, C),
            locate(2.0, 4.9, A),

            line (1.05, 2.5, 1.25, 2.5),
            line (2.75, 2.5, 2.95, 2.5),

      green(line (2.0, 4.5, 2.0, 0.5)),
            locate(2.0, 0.5, D),

            arc (2.0, 4.5, 0.8, 0.8, 90, 115),
            arc (2.0, 4.5, 0.8, 0.8, 65, 90)
)}}}

        &lt;U&gt;Figure&lt;/U&gt;
 &lt;/TD&gt;
 &lt;/TR&gt;
&lt;/TABLE&gt;&lt;/pre&gt;</PRE>