We are given that, between the triangles, two pairs of corresponding sides are proportional. Without knowing anything else, the proportionality of two pairs of sides is insufficient to prove that the triangles are similar.

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. The angles that are congruent are the included angles of their respective sides. By the SAS Similarity Postulate, this is enough to prove that .

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of larger triangle  equal to of that of smaller triangle —or, equivalently, greater.

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. If the angles were the included angles of the triangles, then the SAS Similarity Theorem could be applied to prove that ; however, the two congruent angles are nonincluded, and there is no "SSA" statement that can be applied to prove similarity. Without further information, it cannot be proved that the triangles are similar.

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of smaller triangle  equal to  of that of larger triangle —or, equivalently,  less.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of larger triangle  equal to  of that of smaller triangle —or, equivalently, 25% greater.

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of the smaller triangle  equal to  of that of larger triangle —or, equivalently,   less.

A scalene triangle has three sides of different lengths. The triangle is known to have sides of length 8 and 12, so this eliminates 8 and 12 as the correct choices for the length of the third side.

The sum of the lengths of the two smallest sides must exceed the length of the third side. 4 can be eliminated as a the correct choice, since , violating this condition.

This leaves 6 and 10 as possible answers. For a triangle to be obtuse, it must hold that if are its sidelengths, the greatest of the three,

.

If the length of the third side is 10, setting , we see that

,

violating this condition.

If the length of the third side is 6, setting , we see that

,

satisfying this condition.

This makes 6 the correct choice.

The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:

, where  is the number of sides.

For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:

Solving for  requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:

Solving for  is straightforward: subtract the values of the 3 known angles from both sides:

 is the central angle that intercepts , so 

.

Therefore, we need to find  to obtain our answer.

If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore, 

Letting , since the total arc measure of a circle is 360 degrees, 

We are also given that 

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the measure of  and, consequently, that of .