Answer:
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
The midpoint of the line [tex]\overline{EF}[/tex] is the point that divides [tex]\overline{EF}[/tex] 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 [tex]{}[/tex] Reason
1. m∠BDC ≅ m∠ABD [tex]{}[/tex] 1. Alternate angles theorem
2. m∠DGF ≅ m∠BGE [tex]{}[/tex]2.Vertical angles theorem
3. [tex]\overline{DC}[/tex] = [tex]\overline {AB}[/tex] [tex]{}[/tex] 3. Opposite sides of a parallelogram ABCD
4. [tex]\overline{CF}[/tex] ≅ [tex]\overline{DF}[/tex] [tex]{}[/tex] 4. Definition of midpoint of DC
5. [tex]\overline{CF}[/tex] = [tex]\mathbf{\overline{DF}}[/tex] [tex]{}[/tex] 5. Definition of congruency
6. [tex]\overline{CF}[/tex] + [tex]\overline{DF}[/tex] = DC [tex]{}[/tex] 6. Segment addition property
7. [tex]\overline{CF}[/tex] + [tex]\overline{CF}[/tex] = DC [tex]{}[/tex] 7. Substitution property
8. 2·[tex]\overline{CF}[/tex] = DC [tex]{}[/tex] 8. Addition
9. [tex]\overline{CF}[/tex] = 0.5· [tex]\overline{DC}[/tex] = [tex]\overline{DF}[/tex] [tex]{}[/tex] 9. Division property
Similarly;
10. [tex]\overline{AE}[/tex] = 0.5·[tex]\overline{AB}[/tex] = [tex]\overline{EB}[/tex] [tex]{}[/tex] 10. Division property
11. 0.5· [tex]\overline{DC}[/tex] = 0.5·[tex]\overline{AB}[/tex] [tex]{}[/tex] 11. Multiplication property of equality
12. [tex]\overline{AE}[/tex] = [tex]\overline{EB}[/tex] [tex]{}[/tex] 12. Substitution property
13. ΔDFG ≅ ΔBGE [tex]{}[/tex] 13. Angle-Angle-Side rule of congruency
14. [tex]\overline{FG}[/tex] ≅ [tex]\overline{EG}[/tex] [tex]{}[/tex] 14. CPCTC [tex]{}[/tex]
15. [tex]\overline{FG}[/tex] = [tex]\overline{EG}[/tex] [tex]{}[/tex] 15. Definition of congruency
16. Point G is the midpoint of [tex]\overline{EF}[/tex] [tex]{}[/tex] 17. Definition of midpoint
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