How to find the unknown side if I only know 2 sides and the mesure of one angle of a triangle ? by the way this is not a right-angled triangle two sides are equal
Assume ABC is triangle . In this triangle futher let AC ,BC and included angle between these sides is known.
Then by triangle law `c^2=a^2+b^2-2abCos(C)` , we can calculate third side.
In our example
We need to determine AB ?
`AB^2=AC^2+BC^2-2 AC.BC. cos( C)`
angle between side AC and BC is C in considered triangle.
So as you prove congruency in triangle , those are the minimum information required to define unique triangle.
In special case we define triangle like as equilateral,isosceles or right angle triangle etc.
For arbitrary triangle we can not define unique triangle.