SAT Mathematics Practice Test 3

Since the angles marked $\mathit{y}°$ and $\mathit{u}$° are vertical angles, $\mathit{y}=\mathit{u}$.
Subtracting the sides of $\mathit{y}=\mathit{u}$ from the corresponding sides of $\mathit{x}+\mathit{y}=\mathit{u}+\mathit{w}$ gives $\mathit{x}=\mathit{w}$.
Since the angles marked $\mathit{w}°$ and $\mathit{z}°$ are vertical angles, $\mathit{w}=\mathit{z}$. Therefore, $\mathit{x}=\mathit{z}$, and so I must be true.
The equation in II need not be true. For example, if $\mathit{x}=\mathit{w}=\mathit{z}=\mathit{t}=70$ and $\mathit{y}=\mathit{u}=40$,
then all three pairs of vertical angles in the figure have equal measure and the given condition $\mathit{x}+\mathit{y}=\mathit{u}+\mathit{w}$ holds.
But it is not true in this case that $\mathit{y}$ is equal to $\mathit{w}$.
Therefore, II need not be true.
Since the top three angles in the figure form a straight angle, it follows that $\mathit{x}+\mathit{y}+\mathit{z}=180$.
Similarly, $\mathit{w}+\mathit{u}+\mathit{t}=180$, and so $\mathit{x}+\mathit{y}+\mathit{z}=\mathit{w}+\mathit{u}+\mathit{t}$.
Subtracting the sides of the given equation $\mathit{x}+\mathit{y}=\mathit{u}+\mathit{w}$ from the corresponding sides of $\mathit{x}+\mathit{y}+\mathit{z}=\mathit{w}+\mathit{u}+\mathit{t}$ gives $\mathit{z}=\mathit{t}.$
Therefore, III must be true. Since only I and III must be true, the correct answer is choice B.
Choices A, C, and D are incorrect because each of these choices includes II, which need not be true.