Given: Quadrilateral ABCD in which AB = DC and AD = BC$$
To prove: ABCD is a parallelogram.
Construction: Draw diagonal AC.
Proof: Some statements and reasons for their validity are given below:
StatementReasonAD=BCGivenAB=CDGivenAC=ACCommon△CBA≅△ADC By SSS postulate∠DCA=∠BACCorresponding angles of congruent trianglesDC∥ABAlternate angles are equal.∠DAC=∠BCACorresponding angles of congruent trianglesAD∥BCAlternate angles are equal.It follows that ABCD is a paralleogram
Answers & Comments
Step-by-step explanation:
Given: Quadrilateral ABCD in which AB = DC and AD = BC$$
To prove: ABCD is a parallelogram.
Construction: Draw diagonal AC.
Proof: Some statements and reasons for their validity are given below:
StatementReasonAD=BCGivenAB=CDGivenAC=ACCommon△CBA≅△ADC By SSS postulate∠DCA=∠BACCorresponding angles of congruent trianglesDC∥ABAlternate angles are equal.∠DAC=∠BCACorresponding angles of congruent trianglesAD∥BCAlternate angles are equal.It follows that ABCD is a paralleogram
Verified answer
A quadrilateral ABCD in which AB=CD and AD=BC.
To prove:- AB∥CD and AD∥BC
Construction:- Join A and C.
Proof:-
In △ABC and △CDA,
AB=CD(Given)
AD=BC(Given)
AC=AC(Common)
∴△ABC≅△CDA(By SSS)
Therefore, by CPCTC,
∠ACD=∠BAC
∠DAC=∠ACB
Since alternate anglea are equal, thus the ines are parallel.
Therefore,
AB∥CD and AD∥BC
Since both the pairs of opposite sides of quadrilateral are parallel, ABCD is a parallelogram.