You can also solve the general case with the two angles being X and Y, and solving for Y: Case 1: X = Y. Case 2: Y is the vertex angle. Y is therefore 180 - 2X. Case 3: X is the vertex angle. Y is therefore (180 - X)/2. Add these up, and you get 270 - 3X/2. To solve this specific case, plug in 70 for X and get 165.
For the third case, where X is equal to 70° i wouldn't have solved this way, but rather with a similar approach to what you did in triangle 2. So for triangle 1, we have 70 and x=55 (70 + 55 + 55) for triangle 2, we have 70 and x=40 (70 + 40 + 70) and for triangle 3, we'd have 70 and x=70 (70 + 70 + 40) The question states that there are two different angles so we can't just take the one angle we know as the one we're looking for (that's what you did with you third triangle and i think it's wrong) but we can totally have two different angles that have the same measure, which is the case if we have an isosceles triangles with two 70° angles. The answer would still be 70° as you said but for a different reason.
The problem statement identifies two angles, 70 and X. In your third example, the angles are 70-55-55, so "70 and X," which by the problem statement must be two different angles, can only be 70 and 55 (same as your first example). For X to also be 70, there must be two 70s, as in your middle example 70-70-40. So in total, there are only two values for X. (70, X=55, 55 and 70, X=70, 40)
You can also solve the general case with the two angles being X and Y, and solving for Y:
Case 1: X = Y.
Case 2: Y is the vertex angle. Y is therefore 180 - 2X.
Case 3: X is the vertex angle. Y is therefore (180 - X)/2.
Add these up, and you get 270 - 3X/2. To solve this specific case, plug in 70 for X and get 165.
For the third case, where X is equal to 70° i wouldn't have solved this way, but rather with a similar approach to what you did in triangle 2.
So for triangle 1, we have 70 and x=55 (70 + 55 + 55)
for triangle 2, we have 70 and x=40 (70 + 40 + 70)
and for triangle 3, we'd have 70 and x=70 (70 + 70 + 40)
The question states that there are two different angles so we can't just take the one angle we know as the one we're looking for (that's what you did with you third triangle and i think it's wrong) but we can totally have two different angles that have the same measure, which is the case if we have an isosceles triangles with two 70° angles. The answer would still be 70° as you said but for a different reason.
The problem statement identifies two angles, 70 and X. In your third example, the angles are 70-55-55, so "70 and X," which by the problem statement must be two different angles, can only be 70 and 55 (same as your first example). For X to also be 70, there must be two 70s, as in your middle example 70-70-40. So in total, there are only two values for X. (70, X=55, 55 and 70, X=70, 40)
Poorly phrased. For the 3rd triangle, you don't have TWO angles which are 70 AND x, you have ONE angle which equals 70 AND ALSO x.