Since the triangle is isosceles, and <A is located at the top of the triangle that is on its base, <B and <C are equivalent.  Since <B is 84 degrees, <C is also.  There are 180 degrees in a triangle so 180 - 84 - 84 = 12 degrees.

Since we are given expressions for the two congruent angles of the isosceles triangle, we can set the expressions equal to see how x relates to y. We get,

x – 3 = y – 7 --> y = x + 4

Logically, y must be the greater number if it takes x an additional 4 units to reach its value (knowing they are both positive integers).

We know that an isosceles triangel has two equal sides and thus two equal angles opposite those equal sides. Because there is one obtuse angle of 112 degrees we automatically know that this angle is the vertex. If you sum any triangle's interior angles, you always get 180 degrees. 

180 – 112 = 68 degrees. Thus there are 68 degrees left for the two equal angles. Each angle must therefore measure 34 degrees.

To solve this problem, apply the formula: 

Side , side  must equal 

Since this is an Isosceles triangle, there must be two sides with a length of .

Thus, plug in each side lengths value to find the solution:

We know the angles in a triangle must add up to 180, so we can solve for x. 

3x + 4x + 10 + 8x + 20 = 180

15x + 30 = 180

15x = 150

x = 10

The given triangle is obtuse. Thus, angle  is greater than 90 degrees. A triangle has a sum of 180 degrees, so angle  + angle  + angle  = 180. Since angle C is greater than 90 then angle  + angle  must be less than 90 and it follows that Quantity A is greater. 

To find the top inner angle, recognize that a straight line contains 180^o; thus we can subtract 180 – 115 = 65^o. Since we are given the other interior angle of 30 degrees, we can add the two we know: 65 + 30 = 95^o.

180 - 95 = 85

The question doesn't tell us if this is a right triangle, so we can't assume that it is. But there is a formula to find the area when we don't know the height: area = [p(p – a)(p – b)(p – c)]^1/2, where a, b, and c are the side lengths and p is half of the perimeter. The perimeter is 18 + 24 + 30 = 72, so p = 72/2 = 36.

Then area = [36(36 – 18)(36 – 24)(36 – 30)]^1/2 = [36 * 12 * 6 * 18]^1/2 = 216.

Since we are told nothing about the angles we cannot assume that these are isosceles triangles and are open to possibilites such as that shown below in which the left side would be larger. If both were isosceles triangles then the right side would be larger.

This question draws from the Third Side Rule of triangles. The length of any side of a triangle must be greater than the difference between the other sides and less than the sum of the other two sides.

This means that side x must be between 2 and 8 since the difference between 5 – 3 = 2 and the sum of 3 + 5 = 8.

Choices 3, 4, and 6 all fall within the range of 2 to 8, but choice 9 does not. The answer is 9.