Complete the two-column proof.%0D%0A%0D%0AGiven: %0D%0AProve: x = 78%0D%0A %09a. __________%0D%0A%09b. __________%0D%0A x = 78%09c. ___________
a. Step 1: Statement - Given
Reason - As stated in the problem.
Step 2: Statement - Assume x is equal to 78.
Reason - We assume x to be equal to 78 to begin the proof.
Step 3: Statement - x - 7 = 71
Reason - Subtracting 7 from both sides of the assumption.
Step 4: Statement - Apply the Transitive Property of Equality
Reason - Since both x - 7 = 71 and 71 = 78 are true, we can use the transitive property to state that x - 7 = 78.
Step 5: Statement - x = 85
Reason - Adding 7 to both sides of the equation x - 7 = 78.
Step 6: Statement - There is a contradiction in Step 5.
Reason - Our assumption that x is equal to 78 led to the contradiction that x is equal to 85. Therefore, our assumption was incorrect.
Step 7: Statement - Therefore, x is not equal to 78.
Reason - We have proven that x cannot be equal to 78 based on the contradiction in Step 6.
Step 8: Statement - Converse of the Law of Detachment: If a given value does not satisfy a statement, then that statement cannot be true for that value.
Reason - Applying the converse of the Law of Detachment to our conclusion in Step 7.
Step 9: Statement - Therefore, x = 78 is false.
Reason - Based on the conclusion in Step 8.
Step 10: Statement - x = 78 is not true.
Reason - As stated in Step 9.
Step 11: Statement - x = 78 is false.
Reason - Same as Step 9.