State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a parallelogram. If $AB = 12 \text{ cm}$ and $BC = 5 \text{ cm}$,then $AC = 13 \text{ cm}$.
$(2)$ $ABCD$ is a trapezium. If $AB \parallel CD$ and $AB = 10 \text{ cm}$,then $CD = 10 \text{ cm}$.

(A) $(1)$ False. In a parallelogram $ABCD$,the sides $AB$ and $BC$ are adjacent sides. The diagonal $AC$ forms a triangle $ABC$. By the triangle inequality theorem,the sum of any two sides must be greater than the third side. Here,$AB + BC = 12 + 5 = 17 \text{ cm}$,which is greater than $AC = 13 \text{ cm}$. However,there is no condition that $ABCD$ must be a rectangle. If it were a rectangle,$AC$ would be $\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}$. Since it is only stated as a parallelogram,$AC$ can take various values depending on the angle $\angle B$. Thus,the statement is not universally true.
$(2)$ False. In a trapezium $ABCD$ where $AB \parallel CD$,the parallel sides (bases) do not necessarily have to be equal. If the parallel sides were equal,the figure would be a parallelogram,not necessarily just a trapezium.