Question

m∠R = 60, m∠S = 80, m∠F = 60, m∠D = 40, . Are the two triangles congruent? If yes, explain and tell which segment is congruent to

Answer 1

The correct answer is in file attached  
we have

triangle RST
m∠R = 60, m∠S = 80  and m∠T=180-(80+60)=40 RS=4
triangle EFD
m∠F = 60, m∠D = 40  and m∠E=180-(60+40)=80
EF=4  

Therefore  
The triangles RST  and EFD are congruents because they have two angles and the side common to them, respectively, equal.
This is the theorem of ASA (angle-side-angle).
Explication
 Side common--------> RS=EF
Angles RS------------- > m∠R = 60  m∠S = 80 
Angles EF------------- > m∠E = 80  m∠F = 60   
Angles RS and Angles EF are equals   

The segment which is congruent to RT is FD, because angles of RT are 60 and 40, and angles of FD also are 60 and 40.