Angle-Side-Angle (ASA) congruence rule

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Mathematics Geometry Angle-Side-Angle (ASA) congruence rule

	Angle-Side-Angle (ASA) congruence rule:
	Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle. We are given two triangles ABC and DEF in which: B = E, C = F and BC = EF We need to prove that ? ABC ? DEF. For proving the congruence of the two triangles see that three cases arise. Case 1: Let AB = DE (see Fig. 7.12). Now what do you observe? You may observe that AB = DE (Assumed) B = E (Given) BC = EF (Given) ? ABC ? DEF (By SAS rule) Case 2: Let if possible AB > DE. So, we can take a point P on AB such that PB = DE. Now consider Δ PBC and Δ DEF Observe that in ? PBC and ? DEF, PB = DE (By construction) B = E (Given) BC = EF (Given) So, we can conclude that: ? PBC ? DEF by the SAS axiom for congruence. Since the triangles are congruent, their corresponding parts will be equal. So,PCB = DFE But, we are given that ACB = DFE. So, ACB = PCB. This is possible only if P coincides with A or, BA = ED. So, ? ABC ? DEF (by SAS axiom) Case 3: If AB < DE, we can choose a point M on DE such that ME = AB and repeating the arguments as given in Case 2, we can conclude that AB = DE and so, ? ABC ? DEF. Suppose, now in two triangles two pairs of angles and one pair of corresponding sides are equal but the side is not included between the corresponding equal pairs of angles. We will observe that they are congruent. We know that the sum of the three angles of a triangle is 180°. So if two pairs of angles are equal, the third pair is also equal (180° ? sum of equal angles). So, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. We may call it as the AAS Congruence Rule. RHS Congruence Rule:Two right-angled triangles are congruent if the hypotenuses and one pair of corresponding sides are equal.