Question 165224
Here's a quick note about what the ratio is really saying:


Since the ratio is vertex angle A to the measure of angle B, this means that we have this ratio:


A:B ---> 2:5


So this means that the vertex angle is SMALLER than the base angle. However, if you have a base angle of 40 degrees, this means that the vertex angle is {{{180-2*40=180-80=100}}} which means that the vertex angle is larger (not smaller) than the base angle. So this effectively rules out the "2nd Approach" answer. Remember, the ratio is vertex:base angle (not the other way around)



However, I agree with your first approach. That answer is correct. You'll find that the base angles are 75 degrees and the vertex angle is 30 degrees. Since the ratio of 30:75 reduces to 2:5, this verifies the answer.



Note: it is IMPOSSIBLE to have both an isosceles triangle AND have a base angle that is greater than 90 degrees. Why? Base angles are equal. So if there is an obtuse base angle, there are really 2 obtuse angles in the triangle (which is impossible; triangles can only have at most one obtuse angles). So this means that either the book is wrong or maybe you read the wrong answer.