Question

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Find the measure of one interior angle and one exterior angle in each regular polygon. Round your answer to the nearest tenth if necessary 3aa and 4aa

Answer 1

Answer:

3.

one interior angle: 90°

one exterior angle: 90°

4.

one interior angle: 140°

one exterior angle: 40°

Step-by-step explanation:

Note:
For interior angle:
The sum of the interior angles of a polygon is  $\boxed{\bold{(n-2)*180}}$,  so the measure of one interior angle$\boxed{\bold{\frac{(n-2)*180}{n}}}$ .where n is the number of sides. A regular polygon has all of its interior angles equal.

For exterior angle:

The sum of the exterior angles of a polygon is 360°, so the measure of one exterior angle is  $\boxed{\bold{\frac{360}{n}}}$.

The exterior angle of a polygon is supplementary to its interior angle, meaning that the two angles add up to 180 degrees.

Now for the question:

3.

no. of side (n)=4

one Interior angle: by substituting the value of n

we get,

$\boxed{\bold{\frac{(n-2)*180}{n}=\frac{(4-2)*180}{4}=90^o}}$

one exterior angle: by substituting the value of n

we get,

$\boxed{\bold{\frac{360}{4}=90^o}}$

4.

no. of side (n)=9

one Interior angle: by substituting the value of n

we get,

$\boxed{\bold{\frac{(n-2)*180}{n}=\frac{(9-2)*180}{9}=140^o}}$

one exterior angle: by substituting the value of n

we get,

$\boxed{\bold{\frac{360}{9}=40^o}}$

Answer 2

Answer:

3)  Interior Angle = 90°

    Exterior Angle = 90°

4)  Interior Angle = 140°

    Exterior Angle = 40°

Step-by-step explanation:

A regular polygon is a polygon that has all sides of equal length and all angles of equal measure.

The interior angle of a regular polygon is the angle formed by two adjacent sides within the polygon.

The formula for the interior angle of a regular polygon is:

$\boxed{\begin{minipage}{6cm}\underline{Interior Angle of a Regular Polygon}\\\\Interior Angle =\dfrac{(n-2)\cdot 180^{\circ}}{n} \\\\where:\\\phantom{ww} \bullet n is the number of sides.\\ \end{minipage}}$

The exterior angle of a regular polygon is the angle formed by one side of the polygon and the extension of the adjacent side.

The sum of the exterior angles of any polygon is 360°.

Therefore, the formula for the exterior angle of a regular polygon is:

$\boxed{\begin{minipage}{6cm}\underline{Exterior Angle of a Regular Polygon}\\\\Exterior Angle =\dfrac{360^{\circ}}{n} \\\\where:\\\phantom{ww} \bullet n is the number of sides.\\ \end{minipage}}$

Question 3

The given regular polygon has 4 sides.

Therefore substitute n = 4 into the formulas to find the measure of one interior angle and one exterior angle.

$\boxed{\begin{aligned}\textsf{Interior Angle}&=\dfrac{(4-2)\cdot 180^{\circ}}{4}\\\\&=\dfrac{2\cdot 180^{\circ}}{4}\\\\&=\dfrac{360^{\circ}}{4}\\\\&=90^{\circ}\end{aligned}}$

$\boxed{\begin{aligned}\textsf{Exterior Angle}&=\dfrac{360^{\circ}}{4}\\\\&=90^{\circ}\end{aligned}}$

Question 4

The given regular polygon has 9 sides.

Therefore substitute n = 9 into the formulas to find the measure of one interior angle and one exterior angle.

$\boxed{\begin{aligned}\textsf{Interior Angle}&=\dfrac{(9-2)\cdot 180^{\circ}}{9}\\\\&=\dfrac{7\cdot 180^{\circ}}{9}\\\\&=\dfrac{1260^{\circ}}{9}\\\\&=140^{\circ}\end{aligned}}$

$\boxed{\begin{aligned}\textsf{Exterior Angle}&=\dfrac{360^{\circ}}{9}\\\\&=40^{\circ}\end{aligned}}$