Proving Congruent Triangles

Two triangles are congruent if their corresponding parts are congruent. We need to check whether corresponding sides and angles are congruent.

Are the Sides Congruent?

From the diagram, we can see that $\overline{TR}\cong \overline{TU}$ and $\overline{RK}\cong \overline{UK}.$ Moreover, $\overline{TK}\cong \overline{TK}$ since any segment is congruent to itself. Therefore, the three sides of $△TRK$ are congruent to the three sides of $△TUK.$

Are the Angles Congruent?

To determine whether the third angles are congruent, we will first use the Triangle Angle Sum Theorem and the Transitive Property of Equality. $$\begin{array}{c}m\angle KRT+m\angle RTK+m\angle TKR=180\\ m\angle KUT+m\angle UTK+m\angle TKU=180\\ \Downarrow \\ m\angle KRT+m\angle RTK+m\angle TKR\\ =\\ m\angle KUT+m\angle UTK+m\angle TKU\end{array}$$ In the given diagram, we see that $\angle KRT$ $\cong$ $\angle KUT$ and that $\angle RTK$ $\cong$ $\angle UTK.$ Therefore, $m\angle KRT$ $=$ $m\angle KUT$ and $m\angle RTK$ $=$ $m\angle UTK.$ Let's use this information to show that $\angle TKR$ $\cong$ $\angle TKU.$ Since $m\angle TKR$ $=$ $m\angle TKU,$ we conclude that $\angle TKR$ $\cong$ $\angle TKU.$

Conclusion

Since all the corresponding parts are congruent, we can conclude that $△RKT$ $\cong$ $△UKT.$