Final answer:
By using the properties of congruent triangles and the commutative property of addition, we can show that AB + DE is greater than AD - BE since congruent triangles have equal corresponding sides, and the sum of positive numbers is greater than their difference.
Step-by-step explanation:
To prove that AB + DE > AD - BE, given that △ABC ≅ △DEC, we start by analyzing the properties of congruent triangles. Since triangles ABC and DEC are congruent, their corresponding sides are equal, which gives us AB = DE and BC = EC.
Utilizing the properties of congruent triangles and the commutative property of addition, we can approach the inequality to be proven.
- Step 1: From triangle congruence, AB = DE and BC = EC.
- Step 2: Add AB + DE on both sides of the first equation.
- Step 3: Using the commutative property of addition (A + B = B + A), we can reorganize to AB + AB > AD - BE. Since AB = DE, the expression simplifies to AB + DE > AD - BE.
- Step 4: The inequality holds true because it's based on the lengths of lines which by definition must be positive, so the sum of two sides must be greater than the difference.
This steps lead us to conclude that the original statement is correct.