Considering the temperature vs. time graph below, how does the temperature at the beginning of a change of state compare with the temperature at the end of the change?

The temperature is always the same. Hence, option A is correct.
The degree of hotness or coldness is measured on a definite scale.
Assume we have something in the solid phase. As we increase the temperature, the particles on the solid increase their kinetic energy, thus, the particles move more.
This causes the volume of the object increases (for example when we heat up the metal and it dilates). This keeps happening until we reach a critical point when we are near a change of phase.
At this point, the energy given is not used to increase the temperature of the object but is used to "break" bonds in such a way that the particles are freer than before. When all these bonds are "broken" the change of phase is completed, and in the case of the solid, we go from solid phase to liquid phase.
An example of this is that we can have liquid water and solid water both at 0°C, so if you have a thermometer in your home, an experiment that you can do is:
Put water in the refrigerator.
Note when the liquid water reaches 0°C
As the water starts to solidify, keep recording its temperature, you will see that it does not change (a lot, it may change a little bit) until all the water changes phase.
Hence, the temperature is always the same.
Learn more about the temperature here:
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