write a 2 column proof, Given D is the midpoint of AC, E is the midpoint of BC and DE is congruent to EF, prove: DE is parallel to AB and DE = 1/2 AB. answer is 15 steps
I'd need to see the figure.
To write a 2-column proof for the given statement, "Given D is the midpoint of AC, E is the midpoint of BC, and DE is congruent to EF, prove: DE is parallel to AB and DE = 1/2 AB," we will need to provide statements and corresponding reasons. Here is a step-by-step guide to constructing the proof:
Step 1: Draw a diagram illustrating the given information, with points A, B, C, D, E, and F.
Step 2: Write down the given information as statements:
Given: D is the midpoint of AC, E is the midpoint of BC, and DE is congruent to EF.
Step 3: Introduce notation: Let DE = EF = x, AB = y.
Step 4: Use the definition of midpoint to write statement 1:
Statement 1: AD = DC.
Step 5: Use the definition of midpoint to write statement 2:
Statement 2: BE = EC.
Step 6: Use the definition of congruent segments to write statement 3:
Statement 3: DE = EF = x.
Step 7: Use the definition of congruent segments to write statement 4:
Statement 4: AB = y.
Step 8: Use the transitive property to write statement 5:
Statement 5: AD = DC = EC.
Step 9: Use the transitive property and statement 5 to write statement 6:
Statement 6: AD = EC.
Step 10: Use the definition of midpoint and statement 6 to write statement 7:
Statement 7: AE || BD.
Step 11: Use the definition of congruent segments and statement 3 to write statement 8:
Statement 8: DE || BF.
Step 12: Use alternate interior angles theorem and statement 7 to write statement 9:
Statement 9: ∠DAE ≅ ∠BDE.
Step 13: Use the corresponding angles theorem and statement 9 to write statement 10:
Statement 10: ∠BDE≅∠FDE.
Step 14: Use the definition of congruent angles and statement 10 to write statement 11:
Statement 11: ∠BDF ≅ ∠FDE.
Step 15: Use the definition of corresponding angles and statement 11 to write statement 12:
Statement 12: DE || AB.
Step 16: Use the definition of midpoint and statement 7 to write statement 13:
Statement 13: DE = 1/2 AD.
Step 17: Use statement 13 and the transitive property to write statement 14:
Statement 14: DE = 1/2 AB.
Finally, conclude the proof with the desired statements:
Conclusion 1: DE is parallel to AB (Statement 12)
Conclusion 2: DE = 1/2 AB (Statement 14)
Remember, the order of the statements can be rearranged to create a logically organized proof.