In parallelogram abcd, e is the midpoint of ab and f is the midpoint of dc . let g be the intersection of the diagonal db and the line segment ef . prove that g is the midpoint of ef.

The midpoint of the line is the point that divides in two halves of the same length.

Reasons:

The given parameters are;

The midpoint of AB in parallelogram ABCD = E

The midpoint of DC = F

Point of intersection of EF and DB = Point G

Required:

To prove that point G is the midpoint of EF.

Solution:

Statement                               Reason

1. m∠BDC ≅ m∠ABD            1. Alternate angles theorem

2. m∠DGF ≅ m∠BGE           2.Vertical angles theorem

3. =                            3. Opposite sides of a parallelogram ABCD

4. ≅                           4. Definition of midpoint of DC

5. =                            5. Definition of congruency

6. + = DC                 6. Segment addition property

7. + = DC                  7. Substitution property

8. 2· = DC                         8. Addition

9. = 0.5· =          9. Division property    

Similarly;

10. = 0.5· =          10. Division property

11. 0.5· = 0.5·             11. Multiplication property of equality

12. =                           12. Substitution property

13. ΔDFG ≅ ΔBGE                 13. Angle-Angle-Side rule of congruency

14. ≅                          14. CPCTC    

15. =                           15. Definition of congruency

16. Point G is the midpoint of  17. Definition of midpoint

Learn more about the midpoint of a line here:

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GF = GE that prove G is the mid-point of EF

Step-by-step explanation:

In the Parallelogram ABCD

∵E is the mid-point of AB

∵F is the mid-point of CD

∵AB = CD opposite sides in the parallelogram

∴EB = DF⇒(1)

∵AB // CD opposite sides in the parallelogram

∴m∠EBD = m∠FDB alternate angles ⇒(2)

∵BD intersects EF at G

∴m∠BGE = m∠DGF vertically opposite angles ⇒(3)

By using (1) , (2) and (3) you can prove:

ΔBGE is congruent to ΔDGF ⇒ AAS

∴GF = GE

∴G is the mid-point of EF