document.write( "Question 122803: Given: Quadrilateral ABCD; Angle A is conguent to Angle C; Angle B is congruent to Angle D\r
\n" ); document.write( "\n" ); document.write( "Prove: Quadrilateral ABCD is a Parallelogram. \n" ); document.write( "
Algebra.Com's Answer #90155 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Given: Quadrilateral ABCD; Angle A is conguent to Angle C; Angle B is congruent to Angle D
\n" ); document.write( "Prove: Quadrilateral ABCD is a Parallelogram.
\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "          D___________________ C\r\n" );
document.write( "          /&               # /\r\n" );
document.write( "         /                  /\r\n" );
document.write( "        /                  /\r\n" );
document.write( "     A /#                &/B\r\n" );
document.write( "       ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ \r\n" );
document.write( "\r\n" );
document.write( "Given: Quadrilateral ABCD\r\n" );
document.write( "       \r\n" );
document.write( "1. mÐA = mÐC\r\n" );
document.write( "   mÐB = mÐD                     They are measures of congruent angles.   \r\n" );
document.write( "\r\n" );
document.write( "mÐA + mÐB + mÐC + mÐD = 360°     The sum of the measures of the interior \r\n" );
document.write( "                                 angles of a polygon with n sides has sum\r\n" );
document.write( "                                 of angles given by the expression\r\n" );
document.write( "                                 \"%28n-2%29%2A180\"°. Quadrilaterals have 4 \r\n" );
document.write( "                                 sides, which means that n=4. Therefore \r\n" );
document.write( "                                 the sum of the interior angles of a  \r\n" );
document.write( "                                 quadrilateral is \"%284-2%29%2A180\"° which\r\n" );
document.write( "                                 equals \"2%2A180\"° or \"360\"°\r\n" );
document.write( "\r\n" );
document.write( "mÐA + mÐB + mÐA + mÐB = 360°     Equals may be substituted for equals. Here\r\n" );
document.write( "                                 we are given that mÐA = mÐC and that \r\n" );
document.write( "                                 mÐB = mÐD, and we made those substitutions\r\n" );
document.write( "                                 of equals for equals. \r\n" );
document.write( "\r\n" );
document.write( "2mÐA + 2mÐB = 360°               mÐA + mÐA = 2AÐ and mÐB + mÐB = 2ÐB and\r\n" );
document.write( "                                 substituting equals for equals.\r\n" );
document.write( "\r\n" );
document.write( " mÐA + mÐB = 180°                Dividing the equation through by 2.\r\n" );
document.write( "(ÐA and ÐB are supplementary)  \r\n" );
document.write( "\r\n" );
document.write( "AD||BC                           When a transversal (AB) cuts two lines\r\n" );
document.write( "                                 (AD and BC), and the angles on the same\r\n" );
document.write( "                                 side of the transversal (ÐA and ÐB) are\r\n" );
document.write( "                                 suplementary, the lines are parallel.\r\n" );
document.write( "\r\n" );
document.write( "m/C + m/B = 180°                 Substituting mÐC for mÐA in equation 2 steps back.     \r\n" );
document.write( "(ÐC and ÐB are supplementary)\r\n" );
document.write( "\r\n" );
document.write( "AB||CD                           When a transversal (BC) cuts two lines\r\n" );
document.write( "                                 (AB and CD), and the angles on the same\r\n" );
document.write( "                                 side of the transversal (ÐC and ÐB) are\r\n" );
document.write( "                                 suplementary, the lines are parallel.\r\n" );
document.write( "\r\n" );
document.write( "quadrilateral ABCD is a parallellogram.\r\n" );
document.write( "                       \r\n" );
document.write( "                                 Because both pairs of opposite sides are\r\n" );
document.write( "                                 parallel and that completes the definition \r\n" );
document.write( "                                 of a parallelogram.   \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );