Answer
The largest angle is $C = 67.4^{\circ}$.
Work Step by Step
1. Find $\angle B$
$cos(B) = \frac{a^{2} + c^{2} -b^{2}}{2ac}$
$cos(B) = \frac{(13)^{2} + (15)^{2} -(14)^{2}}{2(13)(15)}$
$cos(B) = \frac{198}{2(13)(15)}$
$cos(B) = \frac{198}{390}$
$cos(B) = 0.50769...$
by GDC / calculator
$B = cos^{-1}(0.50796...)$
$B = 59.4897...^{\circ}$
$B = 59.5^{\circ}$
2. Find $\angle A$
$cos(A) = \frac{b^{2} + c^{2} -a^{2}}{2bc}$
$cos(A) = \frac{(14)^{2} + (15)^{2} -(13)^{2}}{2(14)(15)}$
$cos(A) = \frac{252}{420}$
$cos(A) = 0.6$
by GDC / calculator
$A = 53.1^{\circ}$
3. Find $\angle C$
$cos(C) = \frac{a^{2} + b^{2} -c^{2}}{2ab}$
$cos(C) = \frac{(13)^{2} + (14)^{2} -(15)^{2}}{2(13)(14)}$
$cos(C) = \frac{140}{364}$
$cos(C) = 0.384615...$
by GDC / calculator
$C = 67.38^{\circ}$
$C = 67.4^{\circ}$
$C \gt B \gt A$
Therefore the largest angle is $C = 67.4^{\circ}$.