For a triangle to be a right triangle, the sides must obey the Pythagorean Theorem. Let's try our options.

3, 4, 5: You should know this is a right triangle without having to do any calculations because it is one of the special triangles that you should remember. But if you didn't, $\displaystyle 3^{2} + 4^{2} = 25 = 5^{2}$.

28, 45, 53:  $\displaystyle 28^{2} + 45^{2} = 784 + 2025 = 2809 = 53^{2}$

45, 55, 75:  $\displaystyle 45^{2} + 55^{2} = 2025 + 3025 = 5050 \neq 75^{2}$. The sides don't follow the Pythagorean Theorem so this can't be a right triangle. This is our answer. Let's check the remaining two sets of sides as well.

48, 64, 80:  $\displaystyle 48^{2} + 64^{2} = 2304 + 4096 = 6400 = 80^{2}$. These are pretty big numbers and this math might take a while. Instead of doing these calculations, we could also see if 48, 64, and 80 look like any of the special triangles we know. Let's divide the three numbers by 16. 48/16 = 3, 64/16 = 4, and 80/16 = 5. Then this is just a type of 3,4,5 triangle, which we know is a right triangle.

84, 35, 91:  $\displaystyle 84^{2} + 35^{2} = 7056 + 1225 = 8281 = 91^{2}$. Again, these are big numbers to square. Let's divide the three numbers by their greatest common factor, 7. 84/7 = 12, 35/7 = 5, 91/7 = 13. Then this is a 5, 12, 13 triangle, which is another of our special triangles that we know is a right triangle.

In order for two right triangles to be similar, the ratio of their dimensions must be equal. First we can check the ratio of the height to the base for the given triangle, and then we can check each answer choice for the triangle with the same ratio:

$\displaystyle Given:\frac{h}{b}=\frac{6}{9}=\frac{2}{3}$

So now we can check the ratio of the height to the base for each answer option, in no particular order, and the one with the same ratio as the given triangle will be a triangle that is similar:

$\displaystyle Option1:\frac{h}{b}=\frac{5}{10}=\frac{1}{2}$

$\displaystyle Option2:\frac{h}{b}=\frac{7}{12}$

$\displaystyle Option3:\frac{h}{b}=\frac{8}{14}=\frac{4}{7}$

$\displaystyle Option4:\frac{h}{b}=\frac{12}{20}=\frac{3}{5}$

$\displaystyle Option5:\frac{h}{b}=\frac{10}{15}=\frac{2}{3}$

The triangle with a height of $\displaystyle 10$ and a base of $\displaystyle 15$ has the same ratio as the given triangle, so this one is similar.

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height $\displaystyle h=4$ and a base $\displaystyle b=12$, the ratio $\displaystyle \frac{h}{b}=\frac{4}{12}=\frac{1}{3}$. The only answer provided with a ratio of $\displaystyle \frac{1}{3}$ is $\displaystyle h=2, b=6$. 

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height $\displaystyle h=5$ and a base $\displaystyle b=20$, the ratio $\displaystyle \frac{h}{b}=\frac{5}{20}=\frac{1}{4}$. The only answer provided with a ratio of $\displaystyle \frac{1}{4}$ is $\displaystyle h=3, b=12$.

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height $\displaystyle h$ of $\displaystyle 7$ and a base $\displaystyle b$ of $\displaystyle 42$,  the ratio $\displaystyle \frac{h}{b}=\frac{7}{42}=\frac{1}{6}$. The only answer provided with a ratio of $\displaystyle \frac{1}{6}$ is $\displaystyle h=5, b=30$.

Corresponding angles of similar triangles are congruent, so Statement 1 alone establishes that $\displaystyle \angle A \cong \angle D$ and $\displaystyle \angle C \cong \angle F$; similarly, Statement 2 alone establishes that $\displaystyle \angle A \cong \angle F$ and $\displaystyle \angle C \cong \angle D$. However, neither statement alone establishes the actual measure of any of the four acute angles. 

Assume both statements are true. From transitivity, it holds that $\displaystyle \angle A \cong \angle D \cong \angle C$. Specifically, the two acute angles of $\displaystyle \bigtriangleup ABC$, one of which is $\displaystyle \angle A$, are congruent. Since $\displaystyle \angle B$ is right, this makes $\displaystyle \bigtriangleup ABC$ and isosceles right triangle, and its acute angles, including $\displaystyle \angle A$, have measure $\displaystyle 45 ^{\circ }$.