Question 1189026
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Since the shown angles are equal, the shown segments inside triangle are bisectors of respective angles.


    +----------------------------------------------------------------------+
    |    In any triangle, bisector of an angle divides the opposite side   |
    |        in the ratio equal to the ratio of adjacent sides.            |
    +----------------------------------------------------------------------+


THEREFORE, in the given triangle  {{{y/z}}} = {{{6/4}}}  and  y + z = 5.

    So, the side of the length 5 is divided in the ratio 6:4.

    It means that y = {{{5*(6/10)}}} = {{{6/2}}} = 3;  z = 5-3 = 2.



Similarly, side of the length 6 is divided in propoprtion  {{{x/w}}} = {{{5/4}}}.

    So, the side of the length 6 is divided in the ratio 5:4.

    It means that  x = {{{6*(5/9)}}} = {{{30/9}}} = {{{10/3}}} = 3 {{{1/3}}},  w = 6 - 3 {{{1/3}}} = 2 {{{2/3}}}.


<U>ANSWER</U>.  x = 3 {{{1/3}}};  y = 3;  z = 2;  w = 2 {{{2/3}}}.
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Solved.