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.

We show Statement 1 alone is sufficient:

If $\displaystyle M$ is the midpoint of $\displaystyle \overline{RE}$, then $\displaystyle MR = MC$. Opposite sides of a rectangle are congruent, so $\displaystyle RT = EC$; all angles of a rectangle, being right angles, are congruent, so $\displaystyle \angle MRT \cong \angle MEC$. This sets up the conditions for the Side-Angle-Side Theorem, and $\displaystyle \Delta MRT \cong \Delta MEC$. Consequently, $\displaystyle MT = MC$, and $\displaystyle \Delta MCT$ is isosceles.

Now, we show Statement 2 alone is sufficient:

If $\displaystyle \angle RTM$, and $\displaystyle \angle ECM$ are congruent, then $\displaystyle \angle MTC$ and $\displaystyle \angle MCT$, being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, $\displaystyle \Delta MCT$ is isosceles.

If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.

If we know that  $\displaystyle \angle B$ is an obtuse angle, however, we can deduce that $\displaystyle \angle A$ and $\displaystyle \angle C$ are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that $\displaystyle \angle B$ has the greatest measure, and that its opposite side, $\displaystyle \overline{AC}$,  is the longest.

Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of $\displaystyle \Delta DEF$; Statement 2 only gives us information about another triangle.

Suppose we assume both statements. Then by similarity,

$\displaystyle \frac{AB}{DE} = \frac{BC}{EF}$.

Since $\displaystyle DE = EF$, then

$\displaystyle \frac{AB}{DE} \cdot DE = \frac{BC}{EF} \cdot EF$, or

$\displaystyle AB = BC$. 

This makes $\displaystyle \Delta ABC$ isosceles.

The longest side of a triangle is opposite the angle of greatest measure.

From Statement 1 alone, we can find two possible scenarios with different answers:

Case 1:

$\displaystyle m \angle B = 20 ^{\circ }, m \angle A = 40 ^{\circ }, m \angle C = 120 ^{\circ }$

Case 2:

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

In both cases, $\displaystyle m \angle A =m \angle B + 20^{\circ }$, but in Case 1, $\displaystyle \overline{AB}$ is the longest side, and in Case 2, $\displaystyle \overline{BC }$ is the longest side.

From Statement 2 alone, however, we know that  $\displaystyle m \angle C > 100^{\circ }$, so $\displaystyle \angle C$ is obtuse and the other two angles are acute. That makes $\displaystyle \overline{AB}$ the longest side.

Assume both statements are true.

By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, $\displaystyle \overline{AC} \ncong \overline{BC}$. Statement 2 alone establishes that $\displaystyle \overline{AB} \ncong \overline{BC}$ . However, the two statements together do not establish whether or not $\displaystyle \overline{AB} \ncong \overline{AC}$, so it is not clear whether $\displaystyle \Delta ABC$ is scalene or isosceles.

By definition, a scalene triangle has three noncongruent sides.

Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side $\displaystyle \overline{AB}$ is congruent to either of the other sides.

Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that $\displaystyle \overline{AB} \cong \overline{AC}$. The triangle cannot be scalene.

By definition, a scalene triangle has three noncongruent sides. 

If $\displaystyle AB = 7, AC = 5, BC = 6$, then $\displaystyle AB > AC$ and $\displaystyle BC > AC$, and the triangle is scalene.

If $\displaystyle AB = 7, AC = 5, BC = 7$, then $\displaystyle AB > AC$ and $\displaystyle BC > AC$, but $\displaystyle AB = BC$, so the triangle is not scalene.

The two statements together are insufficient.