Conditions for the AAS – Angle Angle Side congruence
Two triangles are said to be congruent if two angles and non- included side of the one triangle is equal to the two angles and the non- included side of the other.
Experiment to prove Congruence with AAS:
Draw a ∆LMN with ∠M = 40°, ∠N = 70°, LN = 3 cm.
Also, draw another ∆XYZ with ∠Y = 40°, ∠Z = 70°, XZ = 3cm.
We see that ∠M = ∠Y, ∠N = ∠Z and LN = XZ
Make a trace copy of ∆XYZ and try to make it cover LMN with X on L, Y on M and Z on N. Two triangles cover each other exactly.
Therefore ∆LMN ≅ ∆XYZ
Angle Angle Side (AAS) and Angle Side Angle (ASA) are more or less the same congruence condition.
Worked-out problems on angle angle side congruence triangles (AAS postulate):
1. OB is the bisector of ∠AOC, PM ┴ OA and PN ┴ OC. Show that ∆MPO ≅ ∆NPO.
In ∆MPO and ∆NPO
PM ┴ OM and PN ┴ ON
Therefore ∠PMO = ∠PNO = 90°
Also, OB is the bisector of ∠AOC
Therefore ∠MOP = ∠NOP
OP = OP common
Therefore, ∆MPO ≅ ∆NPO by AAS congruence condition