Question

if the diagonqls of a parralogram is equal then prove it is a rectangle

Answer 1

let the parallelogram be ABCD
therefore AC and BD are diagonals
so AC = BD
in ∆ABC and ∆DCB:
1) AB = DC (opposite sides of parallelogram are equal)
2) BC = CB (common side)
3) AC = DB (diagonals are equal)
therefore ∆ABC congruent to ∆DCB (SSS test)
therefore <ABC = <DCB (CPCT) [i]
also <ABC = 180° - <DCB (adjacent angles of parallelogram are supplementary) [ii]
from [i] and [ii]:
<ABC = 180° - <ABC
2<ABC = 180°
<ABC = 90°
therefore <DCB = 90° (from [i])
and <ADC = 90° (opposite angles of parallelogram are equal)
and <DAB = 90° (opposite angles of parallelogram are equal)
therefore <A = <B = <C = <D = 90°
therefore the parallels is a rectangle (as all angles are 90°)