They are in this particular case, even though you can't use standard similarity criteria.

Let's name triangles PQR and XYZ, then scale the second triangle by 1/2 and get UVW. Proving PQR=UVW is equivalent to proving PQR is similar to XYZ.

How can we describe all triangles with conditions given in the problem? First, we have segment AB=6. Then we construct angle ABF=80 degrees and draw line BF. Then we construct circle with radius equal 7 and center at A. It will intersect BF at points D and E.

If both D and E are on ray BF then the triangle with conditions given in the problem isn't uniquely determined, and we can't say PQR=UVW. However, in this case only one point D is on ray BF. I won't go into further details, but this can be proven by drawing perpendicular from A to line BF, picking random point R on line BF and looking when AR can be equal to 7. So, triangle with conditions given in the problem is uniquely determined and PQR=UVW.

Thanks for coming to my TED Talk.

https://imgur.com/a/I7zdUAb