Question

The midpoint of the bottom side of a square is joined to. the ends of the
top side and extended by the same length. The ends of these lines are
joined and perpendiculars are drawn from these points to the bottom
side of the square extended:
1.
Prove that the quadrilateral ob-
tained thus is also a square.​

Answer 1

Let a square ABCD in which L,M,N&O are the midpoints .

in triangle AML and triangle CNO 

      AM = CN ( AB = DC and M and O are the midpoints )

      AL = CM ( AD = BC and L and   N are the midpoints )

      angle MAL = angle NCO ( all angles of a square = 90 degree )

     by AAS critaria

       triangle AML CONGRUENT to triangle CNO

   therefore ML = ON  ( CPCT  )

similarly in triangle MBN CONGRUENT to  LDO  and 

   AND triangle  AML is CONGRUENT  to triangle

now , 

 in Triangle AML ,

 angle AML = angle ALM ( AM = AL ) 

                  = 45 degree

  similarly in triangle LDO 

  angle DLO = 45 degree

 there fore ,

 angle MLO = 90 degree 

by the properties of SQUARE 

 all sides are equal and angles are 90 degree