SOLUTION:
Recall:
- The sum of all exterior angles of any convex polygon is equal to [tex]360^{\circ}.[/tex]
- The sum of all interior angles of a convex n-sided polygon is given by the formula: [tex] (n - 2)180^{\circ} [/tex]
For number 4:
[tex] \small m^{\circ} + 2m^{\circ} + 3m^{\circ} + 4m^{\circ} + 5m^{\circ} + 6m^{\circ} + 7m^{\circ} + 8m^{\circ} = 360^{\circ} [/tex]
Solving for [tex] m,[/tex]
[tex] 36m^{\circ} = 360^{\circ} [/tex]
[tex] m = \dfrac{360}{36} [/tex]
[tex]\boxed{m = 10} [/tex]
For number 5:
Let [tex]2x[/tex] and [tex] 3x[/tex] be the measures of the other two angles of the pentagon.
[tex] 105^{\circ} + 135^{\circ} + 120^{\circ} + 2x + 3x = (5 - 2)(180^{\circ}) [/tex]
[tex] 360^{\circ} + 5x = 540^{\circ} [/tex]
[tex] 5x = 540^{\circ} - 360^{\circ} [/tex]
[tex] 5x = 180^{\circ} [/tex]
[tex] x = \dfrac{180^{\circ}}{5} [/tex]
[tex] x = 36^{\circ} [/tex]
Substituting the value of x, we get
[tex] 2x = 2(36^{\circ}) = \boxed{72^{\circ}} [/tex]
[tex] 3x = 3(36^{\circ}) = \boxed{108^{\circ}} [/tex]
Thus, the other two angles measure 72° and 108°.
