Question 122803
 Given: Quadrilateral ABCD; Angle A is conguent to Angle C; Angle B is congruent to Angle D 
Prove: Quadrilateral ABCD is a Parallelogram. 
<pre><b>

          D___________________ C
          /&               # /
         /                  /
        /                  /
     A /#                &/B
       ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 

Given: Quadrilateral ABCD
       
1. m<font face = "symbol">Ð</font>A = m<font face = "symbol">Ð</font>C
   m<font face = "symbol">Ð</font>B = m<font face = "symbol">Ð</font>D                     They are measures of congruent angles.   

m<font face = "symbol">Ð</font>A + m<font face = "symbol">Ð</font>B + m<font face = "symbol">Ð</font>C + m<font face = "symbol">Ð</font>D = 360°     The sum of the measures of the interior 
                                 angles of a polygon with n sides has sum
                                 of angles given by the expression
                                 {{{(n-2)*180}}}°. Quadrilaterals have 4 
                                 sides, which means that n=4. Therefore 
                                 the sum of the interior angles of a  
                                 quadrilateral is {{{(4-2)*180}}}° which
                                 equals {{{2*180}}}° or {{{360}}}°

m<font face = "symbol">Ð</font>A + m<font face = "symbol">Ð</font>B + m<font face = "symbol">Ð</font>A + m<font face = "symbol">Ð</font>B = 360°     Equals may be substituted for equals. Here
                                 we are given that m<font face = "symbol">Ð</font>A = m<font face = "symbol">Ð</font>C and that 
                                 m<font face = "symbol">Ð</font>B = m<font face = "symbol">Ð</font>D, and we made those substitutions
                                 of equals for equals. 

2m<font face = "symbol">Ð</font>A + 2m<font face = "symbol">Ð</font>B = 360°               m<font face = "symbol">Ð</font>A + m<font face = "symbol">Ð</font>A = 2A<font face = "symbol">Ð</font> and m<font face = "symbol">Ð</font>B + m<font face = "symbol">Ð</font>B = 2<font face = "symbol">Ð</font>B and
                                 substituting equals for equals.

 m<font face = "symbol">Ð</font>A + m<font face = "symbol">Ð</font>B = 180°                Dividing the equation through by 2.
(<font face = "symbol">Ð</font>A and <font face = "symbol">Ð</font>B are supplementary)  

AD||BC                           When a transversal (AB) cuts two lines
                                 (AD and BC), and the angles on the same
                                 side of the transversal (<font face = "symbol">Ð</font>A and <font face = "symbol">Ð</font>B) are
                                 suplementary, the lines are parallel.

m/C + m/B = 180°                 Substituting m<font face = "symbol">Ð</font>C for m<font face = "symbol">Ð</font>A in equation 2 steps back.     
(<font face = "symbol">Ð</font>C and <font face = "symbol">Ð</font>B are supplementary)

AB||CD                           When a transversal (BC) cuts two lines
                                 (AB and CD), and the angles on the same
                                 side of the transversal (<font face = "symbol">Ð</font>C and <font face = "symbol">Ð</font>B) are
                                 suplementary, the lines are parallel.

quadrilateral ABCD is a parallellogram.
                       
                                 Because both pairs of opposite sides are
                                 parallel and that completes the definition 
                                 of a parallelogram.   

Edwin</pre>