opposite sides and opposite angles of a parallelogram are congruent

 Theorem: opposite sides and opposite angles of a parallelogram are congruent.

GIVEN:▢ ABCD is a parallelogram. it means sideAB ∥SideDC, sideAD∥ sideBC.

TO PROVE :segAD  ≅ segBC, SegDC ≅  segAB,

∠ADC≅ ∠ CBA, and  ∠DAB ≅  ∠ BCD.

CONSTRUCTION :Draw diagonal AC

PROOF:segDC∥segAB  and diagonal AC is transversal.

∴∠ DCA ≅ ∠BAC             ........ ( alternative angles)... (1)

segDA ∥ segBC  and diagonal AC is transversal

  ∴ ∠ DAC  ≅   ∠BCA    ......    ( alternate angles)......(2)

Now,  in   △ ADC and  △ CBA

  ∠DAC ≅   ∠BCA    ......... [from( 2)]

segAC  ≅segCA  .... ( common side)

  ∠DCA ≅   ∠ BAC .....[from(1)]

∴  △  ADC  ≅△CBA   ..... (ASA test of congruence)

 ∴  sideAD≅ sideCB       ...... (c. s.c. t.)

and side DC  ≅ sideAB ..... (c. s. c. t)

Also  ∠ ADC ≅    ∠CBA  ......... (c. a. c. t)

similarly, we can prove  ∠ DAB  ≅  ∠ BCD