Statement 1 is true for any triangle  by the Triangle Inequality, which states that the sum of the lengths of any two sides is greater than that of the third. Therefore, Statement 1 provides unhelpful information.

Statement 2 alone, however, proves that  is obtuse, since the sum of the squares of the lengths of two sides exceeds the square of the length of the third. 

Knowing only the triangle is obtuse only tells you that there is one obtuse angle, but along with the fact that there is a  angle, this allows no further conclusions.

Knowing only that the triangle is isosceles, you can deduce from the Isosceles Triangle Theorem that there are two angles of equal measure; as the measures of the three angles are , there are two possibilities: the triangle is a  triangle, or it is a  triangle, but you cannot choose between the two without further information.

Knowing both facts allows you to choose the first of those two options.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

Statement 1:  Two given lengths with an inscribed angle.

Draw a picture of the scenario.  The values of , , and angle  are known values. 

Use the Law of Cosines to determine side length .

Statement 2:  Two known angles.

There is insufficient information to solve for the length of the hypotenuse with only two interior angles.  The third angle can be determined by subtracting the 2 angles from 180 degrees.

The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles.  There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines. 

Therefore:

Find the length of the hypotenuse of obtuse triangle TLC:

I) 

II) Side T is  

Using I), we can find the measure of all 3 angles:

Next, use II) and the Law of sines to find the hypotenuse:

And we needed both statements to find it!

Since this is an obtuse triangle, pythagorean theorem does not apply.

Statement 1 by itself will only determine a range of values c utilizing the 3rd side rule of triangles. .  Therefore, statement 1 alone is insufficient.

Statement 2 by itself will determine that c is either 10 or 11. Therefore, statement 2 alone is insufficient.

When taken together, statements 1 and 2 define a definitive value for c: . Therefore, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Since  , , , therefore,  thus making this an acute triangle. Pythagorean theorem will not apply.

With the information in statement 1, we can't determine the lengths of any other sides. Therefore, Statement 1 alone is not sufficient.

With the information in statement 2, we can't determine the lengths of any other sides. Therefore, Statement 2 alone is not sufficient.

Using the Third Side Rule for triangles, the information in statements 1 and 2 together would allow us to determine the range of values for c. , but this does not provide a definitive value for c. Therefore, Both statements together are not sufficient.

Therefore - the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

It can already be ascertained from the figure that  , since the left portion is a rectangle, so Statement 2 is redundant.

We can already calculate the area of the rectangular portion of the arrow:

All this is left is to calculate the area of the triangular portion. If we know Statement 1, we can take half the product of the height, which is 13, and the base, which is :

Add these numbers to get the area of the arrow:  

The triangle has as its base a vertical line of length 7, so the height of the triangle would be the perpendicular - which in this case is horizontal - distance from that base. Since the base is part of the -axis, this distance is the absolute value of the -coordinate, which is ony given by Statement 1. Statement 2 is irrelevant.

This is illustrated by this diagram:

The area of a triangle is half the product of its base and its height.

The triangle has as its base a horizontal line of length 10, so the height of the triangle would be the perpendicular - which in this case is vertical - distance from the third vertex to that base. Since the base is part of the -axis, this height is the absolute value of the -coordinate, which is only given by Statement 2. Statement 1 turns out to be irrelevant.

This is illustrated by this diagram:

For a triangle to be isosceles, two of the sides must be equal.  To determine wheter this is true, we must have the three side lengths.  Statement 2 gives us those three side lengths.  However, Statement 1 also gives us all of the information we need by giving us the three vertices.  By using the distance formula, we can easily get the three triangle sides from the vertices.  Therefore both statements alone are sufficient.