Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent.
AB = DE, BC = EF, CA = DF
The triangles are congruent by SSS condition. All three sides of one triangle equal the three sides of the other triangle. ΔABC ≅ ΔDEF.
AB = EF, ∠A = ∠E, AC = ED
The triangles are congruent by SAS condition. Two sides and the included angle are equal. ΔABC ≅ ΔEFD (note: vertices arranged to show correspondence).
AB = DF, ∠B = ∠D = 90°, AC = FE
The triangles are congruent by RHS condition. Both are right-angled triangles with equal hypotenuse and one equal side. ΔABC ≅ ΔDFE.
∠A = ∠D, ∠B = ∠E, AC = DF
The triangles are congruent by AAS condition. Two angles and a non-included side are equal. ΔABC ≅ ΔDEF.
AB = DF, ∠B = ∠F, AC = DE
The triangles are NOT necessarily congruent. This is the SSA condition which does not guarantee congruence. The equal angle is not included between the equal sides.
Explanation
(a) The SSS condition states that if two triangles have the same side lengths, they are congruent. Here AB=DE, BC=EF, and CA=DF, so all corresponding sides are equal. (b) The SAS condition requires two sides and the included angle. Here AB=EF, ∠A=∠E, and AC=ED. The angle is included between the two given sides. (c) The RHS condition applies to right-angled triangles. Here ∠B=∠D=90°, hypotenuse AC=FE, and side AB=DF. This guarantees congruence. (d) The AAS condition states that two angles and a non-included side being equal guarantees congruence. Here two angles are equal and side AC=DF. (e) The chapter explains that SSA (Side Side Angle) condition does not guarantee congruence when the angle is not included between the two sides. Non-congruent triangles can be constructed with these measurements.