(28). In ∆RST the measure of angle m∠S is 125°. Option c is correct.
(29). FG must be greater than 2 and less than 12. Option b is correct.
(28). The sum of interior angles of a triangle is equal to 180°, so for the triangle ∆RST,
x + 16 + 7x + 6 + x + 5 = 180°
9x + 27 = 180
9x = 180 - 27 {collect like terms}
9x = 153
x = 153/9 {divide through by 9}
x = 17
putting the value of x for m∠S = 17x + 6, we have;
m∠S = 7(17) + 6 = 125°
(29). According to the Triangle Inequality Theorem, in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Applying this theorem to triangle FGH, FH + HG must be greater than FG. So, 5 + 7 > FG, which implies FG must be greater than 12, the minimum sum of FH and HG.
Similarly, FH + FG must be greater than HG. So, 5 + FG > 7
FG > 7 - 6
FG > 2
Combining these, we get the range: 2 < FG < 12.
Therefore, in the triangle RST, the angle m∠S= 125°. FG must be greater than 7 and less than 12 based on the Triangle Inequality Theorem.