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
To write a 2-column proof, we need to use statements and reasons. Here is a step-by-step guide on how to prove that DE is parallel to AB and that DE is equal to 1/2 AB using 15 steps:
Statement | Reason
------------------------------------|----------------------------------------
1. D is the midpoint of AC | Given
2. E is the midpoint of BC | Given
3. DE is congruent to EF | Given
4. ΔDEC is congruent to ΔFEB | SAS Congruence
5. ⦳1: ∠DEC ≅ ∠FEA | Corresponding angles of congruent triangles
6. ⦳2: ∠CDE ≅ ∠BEF | Corresponding angles of congruent triangles
7. AB is a transversal | Definition of a transversal
8. ∠DEC and ∠CAB are corresponding angles | Definition of corresponding angles
9. ∠DEC ≅ ∠CAB | Corresponding angles postulate (ⳤ)
10. DE || AB | Converse of the corresponding angles postulate (ⳤ)
11. △ADE is similar to △ACB | AA Similarity (ⳤ)
12. DE/AB = AD/AC | Ratio of corresponding sides of similar triangles
13. AD = AC/2 | Definition of midpoint
14. DE/AB = 1/2 | Substitution (from steps 12 and 13)
15. DE = 1/2 AB | Multiplication property of equality
In step 4, we use SAS congruence to show that triangles DEC and FEB are congruent. Steps 5 and 6 illustrate the corresponding angles of the congruent triangles. In step 9, we apply the corresponding angles postulate (ⳤ) to state that ∠DEC is congruent to ∠CAB. By applying the converse of the corresponding angles postulate in step 10, we conclude that DE is parallel to AB. Finally, using the properties of similar triangles, we obtain the ratio DE/AB and simplify it to 1/2 in steps 12 to 14.