use the given information to prove the following...please help?

Question

Given: AD is congruent to BC
DB is congruent to CA
Prove: Quadrilateral ABCD is an isosceles trapezoid.


i cant put the drawing here but it is a trapezoid with e in the middle as a vertical angle...i think...AB are parrallel to DC...i think...please help me

Answer 1

From the ref. "A trapezoid, in which the non-parallel sides are equal in length, is called isosceles." So what you have given for information, including AB||CD, is an isosceles trapezoid by definition. However, let's at least prove that it's symmetrical in the way we expect for such a figure, namely that the midpoints of the two parallel sides lie on a line perpendicular to those sides.
If A-B-C-D is the sequence of corners encountered in a unidirectional (CW or CCW) tour of the perimeter of the quad, and AB||CD, then assume AB and CD are horizontal and AB < CD and AB is above CD, like this:

......A----------B
.../......................\
D-------------------C

There is no assumption that the midpoint of AB is directly above that of CD but if that is true the quad is isosceles.. Since AB||CD, altitude h of quad is constant. Drop an altitude from B intersecting CD at E. Do the same from A intersecting CD at F.

......A----------B
.../...|..............|...\
D---F---------E---C

You have formed two right triangles BEC and AFD with equal hypotenuses AD and BC and another two sides equal, BE=AF=h. From the Pythagorean theorem the remaining two sides CE and DF are equal . (This is also known as the "hypotenuse-leg" or HL criterion of congruence.) CE=DF means that the midpoint of AB is directly above that of CD and the trapezoid is isosceles.