Why is A.S.S.rule not provided?

Asked by adidas | 6th Mar, 2008, 10:03: AM

Expert Answer:

The SSA (Side-Side-Angle) or ASS (Angle-Side-Side) condition does not guarantee congruence, because it is possible to have two incongruent triangles that satisfy the SSA conditions (two equal corresponding sides and an equal non-included angle). This is known as the ambiguous case. Specifically, SSA fails when the angle is acute, and the side opposite to the angle is shorter than the adjacent side, and the opposite side is longer than the adjacent side times the sine of the angle. In all other cases, SSA is valid.

Thus, the SSA condition does prove congruence when the angle is a right angle. This is known as the HL (Hypotenuse-Leg) condition, or the RHS (Right Angle-Hypotenuse-Side) condition. This is true because the hypotenuse of a right triangle is always longer than either leg.

The SSA condition is also valid if the angle is obtuse; or if the opposite side equals the adjacent side times the sine of the angle (in which case it is a right triangle). (For comparison notice that the opposite side cannot be smaller than the adjacent side times the sine of the angle as the triangle will not "close".)

Answered by  | 7th Mar, 2008, 03:28: PM

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