The longest side is opposite the largest angle for all triangles.

(1) Substituting 3 for $\displaystyle k$ means that $\displaystyle AB=3$ and $\displaystyle BC=5$. But the value of $\displaystyle m$ given for side $\displaystyle AC$ is still unknown $\displaystyle \rightarrow$  NOT sufficient.

 (2) Since $\displaystyle k+3>k$, the longest side must be either $\displaystyle k+3$ or $\displaystyle m$. So, knowing whether $\displaystyle m>k+3$ is sufficient.

If $\displaystyle m=k+4$, knowing that  $\displaystyle k+4>k+3$ , 

then $\displaystyle m>k+3$$\displaystyle \rightarrow$ SUFFICIENT.

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to $\displaystyle 180^{\circ}}$, the only possible angles the other sides could have are $\displaystyle 35^{\circ}$.

Statement 2: This does not provide any information relevant to the question. 

If $\displaystyle \overline{AB} \cong \overline{AC}$, then by the Isosceles Triangle Theorem, $\displaystyle m \angle B = m \angle 2$. Since the sum of the measures of a triangle is 180,

$\displaystyle m \angle A + m \angle B + m \angle 2 = 180$

After some substitution,

$\displaystyle 46 + m \angle 2 + m \angle 2 = 180$

$\displaystyle 46 + 2m \angle 2 = 180$

$\displaystyle 2m \angle 2 = 134$

$\displaystyle m \angle 2 = 67$

Since $\displaystyle \angle 1$ and $\displaystyle \angle 2$ form a linear pair, 

$\displaystyle m \angle 1 + m \angle 2 = 180$, and 

$\displaystyle m \angle 1 = 180 - m \angle 2 = 180 - 67 = 113^{\circ }$

If $\displaystyle m \angle B = 67 ^{\circ }$, then by the Triangle Exterior Angle Theorem, 

$\displaystyle m \angle 2 = m \angle A + m \angle B = 46 + 67 = 113^{\circ }$

So either statement by itself provides sufficient information.

There is an implied condition: $\displaystyle 2x+y=180$. Therefore, with each statement, we have 2 unknown numbers and 2 equations. In this case, we can take a guess that we will be able to find the value of $\displaystyle y$ by using each statement alone. It’s better to check by actually solving this problem.

For statement (1), we can plug $\displaystyle 1.5y$ into $\displaystyle 2x+y=180$. Now we have $\displaystyle 4y=180$, which means $\displaystyle y=45$.

For statement (2), we can rewrite the equation to be $\displaystyle x=112.5-y$ and then plug into $\displaystyle 2x+y=180$, making it

$\displaystyle 2\cdot \left ( 112.5-y \right )+y=180$ 

Then we can solve for $\displaystyle y$ and get $\displaystyle y=45$.

Each statement alone allows us to calculate the measure of one of the angles by subtracting the sum of the other two from 180.

From Statement 1:

$\displaystyle m\angle A + m\angle B + m\angle C = 180^{\circ }$

$\displaystyle 120^{\circ } + m\angle C = 180^{\circ }$

$\displaystyle m\angle C = 60^{\circ }$

From Statement 2:

$\displaystyle m\angle A + m\angle B + m\angle C = 180^{\circ }$

$\displaystyle m\angle A + 110^{\circ } = 180^{\circ }$

$\displaystyle m\angle A =70^{\circ }$

Neither statement alone is enough to answer the question, since either statement leaves enough angle measurement to allow one of the other triangles to be right or obtuse. But the two statements together allow us to calculate $\displaystyle m\angle B$:

$\displaystyle 70 ^{\circ }+ m\angle B + 60^{\circ }= 180^{\circ }$:

$\displaystyle m\angle B = 50 ^{\circ}$

This allows us to prove $\displaystyle \Delta ABC$ acute.

From Statement 2:

$\displaystyle m \angle A +m \angle B + m \angle C = 180^{\circ }$

$\displaystyle 80 ^{\circ }+ m \angle C = 180^{\circ }$

$\displaystyle m \angle C = 100^{\circ } > 90^{\circ}$

This is enough to prove the triangle is obtuse.

From Statement 2 we can calculate $\displaystyle m \angle A$:

$\displaystyle m \angle A + 110 ^{\circ }= 180^{\circ }$

$\displaystyle m \angle A =70 ^{\circ }$ 

We present two cases to demonstrate that this is not enough information to answer the question:

$\displaystyle m \angle A =70 ^{\circ }, m \angle B = 90^{\circ } , m \angle C = 20 ^{\circ}$ - right triangle.

$\displaystyle m \angle A =70 ^{\circ },m \angle B = 80^{\circ } , m \angle C = 30 ^{\circ}$ - acute triangle.

If we assume Statement 1 alone, that $\displaystyle \angle A$ is complementary to $\displaystyle \angle C$, then by definition, $\displaystyle m \angle A + m \angle C = 90 ^{\circ }$. Since $\displaystyle m \angle A+ m \angle B + m \angle C = 180 ^{\circ }$, 

$\displaystyle m \angle B +m \angle A+ m \angle C = 180 ^{\circ }$

$\displaystyle m \angle B +90 ^{\circ } = 180 ^{\circ }$

$\displaystyle m \angle B +90 ^{\circ } -90 ^{\circ } = 180 ^{\circ }-90 ^{\circ }$

$\displaystyle m \angle B =90 ^{\circ }$

This makes $\displaystyle \angle B$ a right angle and $\displaystyle \Delta ABC$ a right triangle.

Statement 2 alone is inufficient, however, since a triangle with exactly two acute angles can be either right or obtuse.

The measure of each of the three angles of the triangle, being angles inscribed in the circle, is one-half the measure of the arc it intercepts. For the triangle to be equilateral, each angle has to measure $\displaystyle 60^{\circ }$, and $\displaystyle m \widehat{AB} = m \widehat{BC} =m \widehat{AC} =120 ^{\circ }$.

Each of the arcs mentioned in the statements is a major arc corresponding to one of these minor arcs, so, specifically, $\displaystyle m \widehat{A C}= 360 ^{\circ } - m \widehat{ABC}$ and $\displaystyle m \widehat{AB} = 360 ^{\circ } - m \widehat{ACB}$. 

From Statement 1 alone, we can calculate:

$\displaystyle m \widehat{A C}= 360 ^{\circ } - m \widehat{ABC} = 360 ^{\circ }- 240 ^{\circ } = 120 ^{\circ }$ 

This does not prove or disprove $\displaystyle \Delta ABC$ to be equilateral, since one minor arc can measure $\displaystyle 120 ^{\circ }$ without the other two doing so.

From Statement 2 alone, we can calculate

$\displaystyle m \widehat{AB} = 360 ^{\circ } - 260^{\circ } = 100 ^{\circ }$

so we know that $\displaystyle \Delta ABC$ is not equilateral.

The measure of each of the three angles of the triangle, being angles inscribed in the circle, is one-half the measure of the arc it intercepts. For the triangle to be equilateral, each angle has to measure $\displaystyle 60^{\circ }$, and $\displaystyle m \widehat{AB} = m \widehat{BC} =m \widehat{AC} =120 ^{\circ }$. This is neither proved nor diproved by Statement 1 alone, since one arc can measure $\displaystyle 120 ^{\circ }$ without the other two doing so; it is, however, disproved by Statement 2 alone.

Statement 1 tells us that the triangle is either right or obtuse, but nothing more.

Statement 2 tells us that the triangle is obtuse. An exterior angle of a triangle is supplemetary to the interior angle to which it is adjacent. Since the supplement of an acute angle is obtuse, this means the triangle must have an obtuse angle.