document.write( "Question 934674: If two non parallel sides of a trapezium are equal, prove that it is cyclic. \n" ); document.write( "

Algebra.Com's Answer #567759 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!
If $\"AD+=+CB\"$ then the trapezoid $\"ABCD\"$ is a $\"cyclic-quadrilateral\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Consider a trapezium $\"ABCD\"$ with $\"AB\"$ || $\"CD\"$ and $\"AD=BC\"$
\n" ); document.write( "Draw AM perpendicular to CD and BN perpendicular to CD.\r
\n" ); document.write( "\n" ); document.write( "In triangle $\"AMD\"$ and triangle $\"BNC\"$ ,
\n" ); document.write( " $\"AD+=+BC\"$ ..................................................... (Given)
\n" ); document.write( "< $\"AMD\"$ = < $\"BNC\"$ ........................................ (By construction, each is $\"90\"$ °)
\n" ); document.write( " $\"AM+=+BM\"$ .................................................... (Perpendicular distance between two parallel lines is same)
\n" ); document.write( " Δ $\"AMD\"$ congruent Δ $\"BNC\"$ ............................................... (RHS congruence rule)
\n" ); document.write( "< $\"ADC\"$ = < $\"BCD\"$ ................................................. (CPCT) ... (1)
\n" ); document.write( "< $\"BAD\"$ and < $\"ADC\"$ .............................................. are on the same side of transversal AD.
\n" ); document.write( "< $\"BAD\"$ + < $\"+ADC\"$ = $\"180\"$ ° ...................................... (2)
\n" ); document.write( "< $\"BAD\"$ + < $\"BCD\"$ = $\"180\"$ °.................................... [Using equation (1)]\r
\n" ); document.write( "\n" ); document.write( "This equation shows that the opposite angles are supplementary.
\n" ); document.write( "Therefore, $\"ABCD\"$ is a $\"cyclic\"$ $\"+quadrilateral\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );