ABCD IS TRAPEIZUM WITH AB PARALLEL. F IS THE MID POINT OF BC. DF AND AB ARE PRODUCED TO MEET AT E. SHOW THAT F IS ALSO THE MID-POINT OF DE.

Question

ABCD IS TRAPEIZUM WITH AB PARALLEL. F IS THE MID POINT OF BC. DF AND AB ARE PRODUCED TO MEET AT E. SHOW THAT F IS ALSO THE MID-POINT OF DE.

Answer 1

I'll assume you mean AB is parallel to CD. Go ahead and draw a picture of the situation as the problem describes it. Our goal is going to be to show that DF and FE are congruent, since that would mean that F is the midpoint of DE.

 

 

1. AB is parallel to CD (given)

 

2. F is the midpoint of BC (given)

 

3. <FDC ≅ <FEB (alternate interior angles theorem, since AB and CD are parallel and DE is a transversal)

 

4. <DFC ≅ <EFB (vertical angles theorem, since the two angles are vertical angles)

 

5. BF ≅ CF (definition of a midpoint, since F is the midpoint of BC)

 

6. ΔFDC ≅ ΔFEB (angle-angle-side, or AAS, since we've shown that two pairs of angles and one pair of sides are congruent in steps 3, 4, and 5)

 

7. DF ≅ EF (corresponding parts of congruent triangles are congruent, or CPCTC, since DF and EF are corresponding sides in our two congruent triangles)

 

8. F is the midpoint of DE (definition of a midpoint, since DF and EF are congruent)