Answer. 90 degrees.


Solution

Let m be the measure of the angle at the base. Then the other angle at the base has the same measure m as the triangle is isosceles.

The vertex angle is 2m according to the condition.

Since the sum of interior angles of a triangle is 189 degrees, it gives you an equation

m + m + 2m = 180,   or  4m = 180.

Hence, m =  = 45 degrees for the angle at the base.

It gives 90 degrees for the vertex angle.

The problem is solved.

Let measure of vertex angle, be V

Then each congruent base angle = , or 

The angles of a triangle sum to , and so, we get: 

V + V = 180

2V = 180

V, or vertex angle measures: , or  

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