Solution to Exercise 2.43.
Definition of a Commonly Used Term in Math
Sketch of a Proof – I could not verify the details, so I will break it down into parts I could not prove.

Exercise 2.43. Justify each numbered step and fill in any gaps in the proof that the Supplement Postulate is not independent of the other axioms.

We list the reasons with an outline of some of the missing steps.

1. Angle Construction Postulate.

2. By the Angle Measurement Postulate, . Hence . Then apply Theorem 2.7.

3. Angle Addition Postulate.

4. Apply the Transitive Property to Steps 1 and 3.

5. We need the conditions for the definition of the interior of an angle. By Step 1, E and C are on the same side of line AB. Since the angles are a linear pair of angles, . Since , by the Crossbar Theorem, A and E are on opposite sides of line BC. Since the angles are a linear pair of angles, A-B-D. Thus A and D are on opposite sides of line BC. Hence, E and D are on the same side of line BC. Therefore, by the definition of the interior of an angle, .

6. Angle Addition Postulate.

7. Angle Measurement Postulate with Steps 4 and 6.

8. Angle Measurement Postulate,  

9. Angle Construction Postulate.

10. Angle Measurement Postulate .

11. Theorem 2.7.

12. Angle Addition Postulate.

13. Substitution with Steps 9 and 12.

14. An argument similar to Step 5. (Write the argument out.)

15. Angle Addition Postulate.

16. Angle Measurement Postulate with Steps 13 and 15.