SAT Mathematics Practice Test 3

Since the angles marked $y^{\circ}$ and $u$° are vertical angles, $y=u$. Subtracting the sides of $y=u$ from the corresponding sides of $x+y=u+w$ gives $x=w$. Since the angles marked $w^{\circ}$ and $z^{\circ}$ are vertical angles, $w=z$. Therefore, $x=z$, and so I must be true.The equation in II need not be true. For example, if $x=w=z=t=70$ and $y=u=40$, then all three pairs of vertical angles in the figure have equal measure and the given condition $x+y=u+w$ holds. But it is not true in this case that $y$ is equal to $w$. Therefore, II need not be true.Since the top three angles in the figure form a straight angle, it follows that $x+y+z=180$. Similarly, $w+u+t=180$, and so $x+y+z=w+u+t$. Subtracting the sides of the given equation $x+y=u+w$ from the corresponding sides of $x+y+z=w+u+t$ gives $z=t.$ 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.