SAT Math: Geometry and Trigonometry

In the figure, line $m$ is parallel to line $n$. What is the value of $w$?

\begin{center} \text{Diagram of 3 lines} \end{center}

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is $10$ feet tall and has a shadow that is $5$ feet long. At the same time, the shadow of the other tree is $2$ feet long. How tall, in feet, is the other tree?

A right triangle ABC is given, with angle ACB being the right angle. The side AB (hypotenuse) has a length of 42, and side AC has a length of 41. What is the value of $\cos A$ in the triangle shown?

In the figure, line $m$ is parallel to line $n$. What is the value of $w$?

Triangle $XYZ$ is similar to triangle $RST$ such that $X$, $Y$, and $Z$ correspond to $R$, $S$, and $T$, respectively. The measure of $\angle Z$ is $20^\circ$ and $2XY = RS$. What is the measure of $\angle T$?

Quadrilateral $P'Q'R'S'$ is similar to quadrilateral $PQRS$, where $P$, $Q$, $R$, and $S$ correspond to $P'$, $Q'$, $R'$, and $S'$, respectively. The measure of angle $P$ is $30^{\circ}$, the measure of angle $Q$ is $50^{\circ}$, and the measure of angle $R$ is $70^{\circ}$. The length of each side of $P'Q'R'S'$ is $3$ times the length of each corresponding side of $PQRS$. What is the measure of angle $P'$?