Troy has written a proof showing that the diagonals of a parallelogram are equal in length. The diagonals of a parallelogram are not necessarily equal so Troy must have made an error. Identify his error(s).
Statement Reason Correct/Incorrect?
A B equals E D Opposite sides of parallelogram blank
angle B A E equals angle A E D Alternate interior angles blank
angle A B D equals angle B D E Alternate interior angles blank
triangle A B C approximately equal to triangle E D C ASA blank
B C equals E C Corresponding sides of congruent triangles blank
A C equals D C Corresponding sides of congruent triangles blank
A E equals B D
A C plus C E equals D C plus C B blank
correct, incorrect, true but based on incorrect statements
The error in Troy's proof lies in statement 4. He states that triangle ABC is approximately equal to triangle EDC using the ASA (Angle-Side-Angle) congruence postulate. However, Troy should have used the SAS (Side-Angle-Side) congruence postulate instead.
Therefore, statement 4 is incorrect, and consequently, statement 5, which claims that BC equals EC based on the congruence of the triangles, is also incorrect. Since the proof relies on these incorrect statements, the final statement is also incorrect.
Troy's errors in his proof are as follows:
1. Statement: A B equals E D
Reason: Opposite sides of parallelogram
Correct/Incorrect? Incorrect
Explanation: The opposite sides of a parallelogram are equal in length, but this statement implies that the diagonal AB is equal to diagonal ED. This is incorrect because the diagonals of a parallelogram are not necessarily equal.
2. Statement: angle B A E equals angle A E D
Reason: Alternate interior angles
Correct/Incorrect? Correct
Explanation: This statement is correct. When a pair of parallel lines is intersected by a transversal, the alternate interior angles formed are congruent.
3. Statement: angle A B D equals angle B D E
Reason: Alternate interior angles
Correct/Incorrect? Incorrect
Explanation: This statement is incorrect. The alternate interior angles formed when a pair of parallel lines is intersected by a transversal are congruent, but in this case, angle ABD and angle BDE are not alternate interior angles.
4. Statement: triangle A B C approximately equal to triangle E D C
Reason: ASA
Correct/Incorrect? True but based on incorrect statements
Explanation: This statement is true, but it is based on incorrect statements. ASA, or Angle-Side-Angle, is a valid congruence criterion for triangles, but angle ABD and angle BDE are not congruent as stated earlier.
5. Statement: B C equals E C
Reason: Corresponding sides of congruent triangles
Correct/Incorrect? Incorrect
Explanation: This statement is incorrect because it is based on the incorrect congruence of triangle ABC and triangle EDC.
6. Statement: A C equals D C
Reason: Corresponding sides of congruent triangles
Correct/Incorrect? Incorrect
Explanation: This statement is incorrect because it is based on the incorrect congruence of triangle ABC and triangle EDC.
7. Statement: A E equals B D
Reason: None given
Correct/Incorrect? Incorrect (missing reason)
Explanation: This statement is incorrect because it is not supported by a reason. Troy should provide a reason or justification for this statement.
8. Statement: A C plus C E equals D C plus C B
Reason: None given
Correct/Incorrect? Incorrect (missing reason)
Explanation: This statement is incorrect because it is not supported by a reason. Troy should provide a reason or justification for this statement.