Write given and to prove.if three sides of triangle are congruent then it's three angles are also congruent
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Congruence in Triangles
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Congruence in two or more triangles depends on the measurements of their sides and angles. The three sides of a triangle determine its size and the three angles of a triangle determine its shape. Two triangles are said to be congruent if pairs of their corresponding sides and their corresponding angles are equal. They are of the same shape and size. There are many conditions of congruence in triangles. Let us discuss them in detail.
Congruence in Triangles
If the three angles and the three sides of a triangle are respectively equal to the corresponding angles and the corresponding sides of another triangle, then both the triangles are said to be congruent. In the Δ PQR and ΔXYZ, as shown below, we can identify that PQ = XY, PR = XZ, and QR = YZ and ∠P = ∠X, ∠Q = ∠Y and ∠R = ∠Z. Then we can say that Δ PQR ≅ ΔXYZ.
congruence in triangles
The two triangles need to be of the same size and shape to be congruent. Both the triangles under consideration should superimpose on each other. When we rotate, reflect, and/ or translate a triangle, its position or appearance seems to be different. In that case, we need to identify the six parts of a triangle and their corresponding parts in the other triangle. Consider Δ ABC and ΔPQR as shown below.
identifying the corresponding parts
Corresponding Vertices A and P, B and Q, C and R
Corresponding Sides AB = PQ, BC = QR, CA = RP
Corresponding Angles ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R
Thus on identifying the corresponding parts of the given triangles, we can confirm that Δ ABC ≅ ΔPQR.
Conditions of Congruence in Triangles
Two triangles are said to be congruent if they are of the same size and same shape. Necessarily, not all the six corresponding elements of both the triangles must be found to be equal to determine that they are congruent. Based on studies and experiments, there are 5 conditions for two triangles to be congruent. They are SSS, SAS, ASA, AAS, and RHS congruence properties.
SSS Criterion for Congruence
SSS Criterion stands for Side-Side-Side Criterion. Under this criterion, two triangles are congruent if three sides of a triangle are equal to the corresponding sides of the other triangle.
SSS Criterion for Congruence
If Δ BAC ≅ ΔXYZ under SSS criterion, then the three angles of ΔBAC are bound to be equal to the corresponding angles of