In each of the following independent situations we want to construct a triangle $ABC$ satisfying the given conditions. In each case state how many such triangles $ABC$ exist up to congruence.
(i) $AB = 30 \quad BC = 95 \quad AC = 55$
(ii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad \angle C = 55 ^ { \circ }$
(iii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad BC = 55$
(iv) $\angle A = 30 ^ { \circ } \quad AB = 95 \quad BC = 55$

We want to construct a triangle ABC such that angle A is $20.21 ^ { \circ }$, side AB has length 1 and side BC has length $x$ where $x$ is a positive real number. Let $N ( x ) =$ the number of pairwise noncongruent triangles with the required properties.
(a) There exists a value of $x$ such that $N ( x ) = 0$.
(b) There exists a value of $x$ such that $N ( x ) = 1$.
(c) There exists a value of $x$ such that $N ( x ) = 2$.
(d) There exists a value of $x$ such that $N ( x ) = 3$.

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$, $AC=AC^{\prime}=2\sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

In $\triangle A B C$, it is known that $\overline { A B } = 4$ and $\overline { A C } = 6$, which is insufficient to determine the shape and size of $\triangle A B C$. However, knowing certain additional conditions (for example, knowing the length of $\overline { B C }$) would uniquely determine the shape and size of $\triangle A B C$. Select the correct options.
(1) If we additionally know the value of $\cos A$, then $\triangle A B C$ can be uniquely determined
(2) If we additionally know the value of $\cos B$, then $\triangle A B C$ can be uniquely determined
(3) If we additionally know the value of $\cos C$, then $\triangle A B C$ can be uniquely determined
(4) If we additionally know the area of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined
(5) If we additionally know the circumradius of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined