Problem 564

We begin by recognizing that there are 6 different symbols, and so each face of the cube contains a different symbol.

From left to right, let us number the views of the cube 1, 2 and 3.

Views 1 and 2 each show a face containing the symbol [[IMAGE0]] .

What symbol is on the face opposite to the face containing [[IMAGE1]]  ?

In view 1, [[IMAGE2]] and [[IMAGE3]] are on faces adjacent to the face containing [[IMAGE4]] , and so neither of these can be the symbol that is on the face opposite [[IMAGE5]] .

In view 2, [[IMAGE6]] and [[IMAGE7]] are on faces adjacent to the face containing [[IMAGE8]] , and so neither of these can be the symbol that is on the face opposite [[IMAGE9]] .

There is only one symbol remaining, and so [[IMAGE10]] must be the symbol that is on the face opposite [[IMAGE11]] , and vice versa.
A net of the cube is shown below.

Isosceles triangles have two equal angles, and so the possibilities for these two triangles are:

1) The two equal angles are each equal to $70\degree$, or 2) The two equal angles are each not equal to $70\degree$. (We note that a triangle can not have three angles measuring $70\degree$ since the sum of the three angles would be $3\times70\degree=210\degree$, which is greater than $180\degree$.) If the two equal angles are each equal to $70\degree$, then the measure of the third angle is $180\degree-2\times70\degree=180\degree-140\degree=40\degree$. If the two equal angles are each not equal to $70\degree$, then the sum of the measures of the two equal angles is $180\degree-70\degree=110\degree$, and so the measure of each of the equal angles is half of $110\degree$ or $55\degree$. We note that in the first triangle, the measure of each of the two remaining angles ($70\degree$ and $40\degree$) is even, and in the second triangle, the measure of each of the two remaining angles ($55\degree$ and $55\degree$) is odd. In the first triangle, the sum of the two equal angles is $S=70\degree+70\degree=140\degree$. In the second triangle, the sum of the two equal angles is $T=55\degree+55\degree=110\degree$. The value of $S+T$ is $140\degree+110\degree=250\degree$.