An acute angle must have measure less than ; both non-right angles of the triangle,  and , must be acute.

Assume Statement 1 alone.  The measure of  must be a multiple of 6 and 7 that is less than 90; however, there are at least two such numbers, 42 and 84. With no way to eliminate either choice, Statement 1 alone provides insufficient information.

Statement 2 alone also provides insufficient information, for similar reasons.  can have measure 24 or 48, for example; the measure of  is 90 minus the measure of , so if  cannot be determined definitively, neither can .

Now, assume both statements to be true. 6 and 7 have LCM 42, so any multiple of both must also be a multiple of 42. There are only two multiples of 42 less than 90 - 42 and 84.

If , then, since the measures of the angles of the acute angles of  a right triangle total , 

48 is a multiple of both 6 and 8, so this scenario is consistent with both statements.

If , then, since the measures of the angles of the acute angles of  a right triangle total , 

6 is not a multiple of 8, so this scenario is inconsistent with Statement 2.

Therefore, it can be determined that .

Assume Statement 1 alone. Let  be the measure of the angle of second-greatest measure. Since the measures form an arithmetic sequence, the three angles measure  for some common difference . Their sum is , so

However, since we do not know that common difference, we cannot determine the other angle measures, or whether the triangle is acute, right, or obtuse. For example, if , the widest of the angles measures ; if , the widest angle measures . In the former case, the triangle is acute, having all of its angles measure less than ; in the latter case, the triangle is obtuse, having an angle that measures greater than .

Statement 2 alone provides insufficient information; a 30-30-120 triangle and a 30-60-90 triangle both fit this condition as well as the conditions of the measures of the angles of a triangle, but the former is obtuse and the latter is right.

Now assume both statements to be true. Then, since one angle measures  by Statement 1, and a second measures , the third measures . This angle is a right angle, so  is a right triangle.

Here, we would need to know the length of at least two sides of the triangle ABC, provided it is a right triangle, in order to calculate the area of ABE.

Statement 1 is insufficient alone because it only tells us a property of the triangle and gives us no information about the lengths of the sides. Similarily, statement 2 alone is insufficient because we can't tell the area of triangle ABE from what is given. We would need the length of the height and the length of EB. 

Statements 1 and 2 taken together are insufficient, because even though the height divides the triangle in two similar triangles, we can't find any ratio to calculate the length of the height AE. Therefore, Statements 1 and 2 taken together are insufficient.

Statement 1) only provides the hypotenuse of the triangle, but it does not imply that both legs of the right triangle are congruent sides.  Of the three interior angles, only the right angle is known with the other 2 unknown interior angles.  

Statement 1) does not have enough information to solve for the area of the triangle.

Statement 2) provides the lengths of both legs.  The formula  can be used to solve for the area of the triangle.

Statement 2) can be used by itself to solve for the area of the triangle.

To find the number of tiles needed, we need to find the area of the lobby and the area of one tile.

I) Gives us the area of one tile.

II) Gives us the length of the hypotenuse of the lobby.

Use II) along with the info given in the question to find the last side (Pythagorean Theorem).

, where SS is the short side, MS is the middle side, and H is the hypotenuse.

From there you can find the area of the lobby.

Use I) along with the area of the lobby to find the number of tiles needed.

Neither statement provides us with sufficient information to answer the question but both statements taken together are sufficient to answer the question.

In order to find the area of the right triangle we need both the height and base. We can use the Pythagorean Theorem to solve for the base. 

Now we can find the area:

(1) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.

If the two shorter sides of a right triangle are equal, that means our other two angles are 45 degrees. This means our triangle follows the ratios for a 45/45/90 triangle, so we can find the remaining sides from the length of the hypotenuse.

I) Tells us we have a 45/45/90 triangle. The ratio of side lengths for a 45/45/90 triangle is .

II) Tells us the length of the hypotenuse. 

Together, we can find the remaining two sides and then the perimeter.

Statement 1: We need additional information.

But this can mean our base and height measure 2 and 24, 4 and 12, or 6 and 8. 

We cannot determine which one based solely on this statement.

Statement 2: We're given the length of the hypotenuse so we can narrow down the possible base and height values. 

We have to see which pair of values makes the statement  true.

The only pair that does is 6 and 8.

We can now find the perimeter of the right triangle: 

 or, if you're more familiar with the equation , then:

Statement 1: In order to find the perimeter of a right triangle, we need to know the lengths of the legs, not the hypotenuse.

Statement 2: Since we have the values to both of the legs' lengths, we can just plug it into the equation for the perimeter: