Question

Two angles of an isosceles triangle measure 69 degrees and x degrees. What is the sum of all possible values of x?

Answer 1

if x and x are the equal angles
69 +2x =180
2x = 111
x = 55.5 degrees

If 69 and x are equal angles, x=69.

if x isthe third angle, x = 180-138 =42
Therefore sum is 69+55.5+42 = 166.5 degrres

Answer 2

In an isosceles triangle there are two equal angles and a different angle. Let the triangle be ABC with angle A and B same and C different.
There are three possibilities:

Case-1:
A = 69
B = x = 69
C = 180 - (69+69) = 180-138 = 42
So one possibility is x = 69

Case-2:
A = 69
B = 69
C = x = 180 - (69+69) = 180-138 = 42
So another possibility is x = 42

Case-1:
C = 69
B = x
A = x
So x + x + 69 = 180
⇒ 2x = 180-69 = 111
⇒ x = 111/2 = 55.5
So another possibility is x = 55.5

Sum of all possible values of x = 69+42+55.5 = 166.5