Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

Steps to Solve

1. Identify Given Measurements

   For Triangle SRT:

   • Angle $S = 44^\circ$
   • Angle $T = 37.4^\circ$
   • Side $SR = 44$
   • Side $ST = 55$

   For Triangle NQP:

   • Angle $Q = 17^\circ$
   • Angle $P = 20^\circ$
   • Side $NQ = 25$
   • Side $NP = 20$

2. Calculate the Missing Angle

   In Triangle SRT, the sum of angles is $180^\circ$. Therefore: [ R = 180^\circ - (S + T) = 180^\circ - (44^\circ + 37.4^\circ) = 98.6^\circ ]

   In Triangle NQP, the missing angle can also be calculated: [ N = 180^\circ - (Q + P) = 180^\circ - (17^\circ + 20^\circ) = 143^\circ ]

3. Check for Angle-Angle Similarity (AA)

   Compare the angles:

   • Triangle SRT angles: $44^\circ$, $37.4^\circ$, $98.6^\circ$
   • Triangle NQP angles: $17^\circ$, $20^\circ$, $143^\circ$

   The angles do not match, so we cannot use AA similarity.

4. Check for Side Length Ratios (SSS)

   Calculate the ratios of corresponding sides: [ \frac{SR}{NQ} = \frac{44}{25}, \quad \frac{ST}{NP} = \frac{55}{20} ] [ \frac{44}{25} \approx 1.76, \quad \frac{55}{20} = 2.75 ]

   The ratios are not equal, so SSS cannot apply.

5. Check for Side-Angle-Side Similarity (SAS)

   Compare one angle with the ratios of the sides forming it: The angle included between the two sides does not correspond well. Therefore, we cannot use SAS.

6. Conclusion

   Since none of the similarity criteria (AA, SSS, SAS) are met, the triangles are Not Similar.