Q: 12     ABCD is a trapezium in which  \small AB\parallel CD and  \small AD=BC  (see Fig. \small 8.23). Show that

 (i) \small \angle A=\angle B

 [Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

            

Answers (1)
S seema garhwal

Given: ABCD is a trapezium in which  \small AB\parallel CD and  \small AD=BC

To prove :\small \angle A=\angle B

Proof:      Let \angle A be \angle1, \angleABC be \angle2, \angle EBC be \angle3, \angle BEC be \angle4.

In AECD,

AE||DC             (Given)

AD||CE             (By cnstruction)

Hence, AECD is a parallelogram.

 AD=CE...............1(opposite sides of a parallelogram)

AD=BC.................2(Given)

From 1 and 2, we get

 CE=BC

In \triangleBCE,

      \angle 3=\angle 4.................3  (opposite angles of equal sides)

       \angle 2+\angle 3=180 \degree...................4(linear pairs)

     \angle 1+\angle 4=180 \degree .....................5(Co-interior angles)

From 4 and 5, we get 

 \angle 2+\angle 3=\angle 1+\angle 4

\therefore \angle 2=\angle 1 \rightarrow \angle B=\angle A          (Since,\angle 3=\angle 4)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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