ABCD is a trapezium in which and AD=BC (see figure). Show that
(iv) diagonal AC=diagonal BD
Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.
ABCD is a trapezium in which we have AD=BC.
By construction, we extend AB and draw a line CE parallel to AD. (1)
We already have because ABCD is a trapezium.
From (1) and (2), we can say that AECD is a parallelogram.
(A quadrilateral is a parallelogram if both the pairs of opposite sides are parallel.)
We have AD=BC (Given) (3)
Also, we have AD=CE (Opposite sides of parallelogram are equal) (4)
From (3) and (4), we can say that BC=CE
(In triangle, angles opposite to equal sides are equal) (5)
We have (Co-Interior angles, ) (6)
And, (Linear pair) (7)
From (6) and (7), we can say that
Using (5) in the above equation, we get
We have showed in solution (i) that (8)
But, we have (Opposite angles in parallelogram are equal) (9)
And, (Alternate Interior angles, ) (10)
Using (9) and (10) in equation (8), we get
Join AC and BD.
Therefore, by SAS congruence rule, we have .
We showed in solution (iii) that
(Corresponding parts of congruent triangles are equal)