Step 1: Recall the cosines formula

            The law of cosines is a formula that relates the three sides of a triangle to

            the cosine of a given angle

              FORMULA: a^2 = b^2 + c^2 - 2bc \cos(A)

                                   b^2 = a^2 + c^2 - 2ac \cos(B)

                                   c^2 = b^2 + a^2 - 2ba \cos(C)

                Where each lowercase letter (like a) is the length of the side opposite the

              vertex labeled with the same capital letter.

Step 2: Determine the largest angle in the triangle

            GIVEN: In a triangle ABC

                                            a  = 9 cm

                                            b  = 10 cm

                                            c  = 13 cm  

                    The largest angle is the one facing the longest side, i.e. C.

                            Substitute the given lengths in the law of cosines formula.

                                     c^2 = b^2 + a^2 - 2ab\cos (C)

                                  13^2 = 10^2 + 9^2 - 2(9*10)\cos (C)

                             \cos (C) = \frac{10^2 + 9^2 - 13^2}{2(90)}

                                \cos C = \frac{100 + 81 - 169}{180}

                               \cos C = \frac{12}{180}

                              \cos C = 0.067

                                     C = \cos^{-1}(0.67)

                                     C = 86.2\degree

                    The largest angle in the triangle = C = 86.2\degree