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4. the measure of the exterior angle of an octagon are m°,2m°,3m°,4m°,5m°,6m°,7m°, and 8m° what is the value of m?
5.three angles of a pentagon are 105°,135° and 120° ,find the other two angles if they are in the ratio 2:3.?​

Sagot :

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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°.