Given ∠ABE = 45° and ∠EAB = 63° in ΔABE and∠MNP= 72° and ∠NMP = 63° in ΔMNP. Are the two triangles, ΔABE and ΔMPN similar? If so, by what criterion?

Answer:Yes ,we can prove the two triangles are similar by angle angle test.Step-by-step explanation:Given:∠ABE = 45° ∠EAB = 63° and∠MNP= 72° ∠NMP = 63°To Prove:ΔABE  ~  ΔMPN Proof:In a Triangle sum of the angles of a triangle is 180° In  ΔMPN∴ ∠MNP +  ∠NMP  + ∠MPN = 180°Substituting the given values we get,[tex]72+63+\angle MPN = 180\\135 + \angle MPN = 180\\\angle MPN = 180-135\\\angle MPN = 45[/tex]∠MPN = 45°  ..........................( 1 )Now,for triangles to be similar minimum two angles should be congruent i.e AA test.all the three sides should be proportional i.e SSS testIn  Δ ABE and Δ  MPN∠ ABE ≅ ∠ MPN = 45°      ……….{From ( 1 ) and Given}∠ EAB ≅ ∠ NMP = 63°     ………...{Given}Δ ABE ~ Δ MPN ….{Angle-Angle test}..........Proved