An isosceles triangle is a triangle which has two sides of equal length.
Since it has two sides of equal length, the isosceles triangle has two equal base angles.
If one base angle is 67.5º then the other base angle has to be 67.5º too.
For any triangle, the sum of all internal angles equals 180º. Let's say the unknown angle is x, then:
x + 67.5º + 67.5º = 180º
Let's solve for x:
x = 180º - 67.5º - 67.5º
x = 180º - 135º
x = 45º
Answer: the measure of the vertex angle is 45º.