Chapter 4.3, Problem 26PPS

To determine

To prove: The congruence of triangles is transitive.

Explanation of Solution

Prove:

Let the triangle $\Delta ABC$ is congruent to the triangle $\Delta PQR$ .

So, the sides of the triangles $\overline{AB}\cong \overline{PQ},\overline{BC}\cong \overline{QR}\text{and}\overline{AC}\cong \overline{PR}$ . Similarly, the angles $\angle A\cong \angle P,\angle B\cong \angle Qand\angle C\cong \angle R$ .

Let consider the triangle $\Delta XYZ$ is also congruent to the triangle $\Delta PQR$ .

So, the sides of the triangles $\overline{XY}\cong \overline{PQ},\overline{YZ}\cong \overline{QR}\text{and}\overline{XZ}\cong \overline{PR}$ . Similarly, the angles $\angle X\cong \angle P,\angle Y\cong \angle Qand\angle Z\cong \angle R$ .

Therefore, by transitive property $\Delta ABC$ is congruent to $\Delta XYZ$ .

Thus, congruence of triangles is transitive.

Hence, proved.