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Given: Parallelogram CDEF; S and T are midpoints of EF and ED.
Prove: SR ≅ FD
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: Parallelogram CDEF
Reason: Given item.
2. Statement: S and T are the midpoints of EF and ED.
Reason: Given item.
3. Statement: FS ≅ SE
Reason: Point S bisects the line segment EF into two equal parts.
4. Statement: ET ≅ TD
Reason: Point T bisects the line segment ED into two equal parts.
5. Statement: FE ║ CD
Reason: The opposite sides of a parallelogram are parallel.
6. Statement: ∠ SET ≅ ∠ TDR
Reason: If a transversal line (ED) passed the two parallel lines (EF and CD), then the alternating interior angles are congruent.
7. Statement: ∠ STE ≅ ∠ DTR
Reason: Vertical angles are congruent.
8. Statement: ∆SET ≅ ∆DTR
Reason: Angle Side Angle (ASA) Postulate.
9. Statement: DR ≅ SE
Reason: Since ∆SET ≅ ∆DTR, then all sides of a triangle are congruent to all sides of other triangle.
10. Statement: DR ≅ SE ≅ FS
Reason: Transitive property of congruence.
11. Statement: FS ║ DR
Reason: Since Point R is colinear with CD and CD is parallel to EF, then it follows that FS is parallel to DR since Point S is colinear with EF.
12. Statement: FDRS is a parallelogram.
Reason: Since FS and DR are parallel and congruent, then it follows that the figure formed by the points FDRS is a parallelogram.
13. Statement: FD ≅ SR
Reason: The two opposite sides of a parallelogram are congruent and parallel.