Question

1. find the measure of a single exterior angle of a square. if necessary, round to the nearest tenth.
2. find the measure of a single exterior angle of the regular polygon shown below. if necessary, round to the nearest tenth.
3. find the measure of a single exterior angle of the regular polygon shown below. if necessary, round to the nearest tenth.
4. the exterior angle of a regular polygon measures $40^{\circ}$. how many sides does the polygon have?
5. the exterior angle of a regular polygon measures $4^{\circ}$. how many sides does the polygon have?
6. the measures of the exterior angles of a hexagon are $x^{\circ}$, $2x^{\circ}$, $4x^{\circ}$, $5x^{\circ}$, $8x^{\circ}$, and $10x^{\circ}$. solve for $x$.
7. the measures of the exterior angles of a triangle are $5x^{\circ}$, $9x^{\circ}$, and $10x^{\circ}$. find the measure of the smallest exterior angle.

Explanation:

Step1: Use exterior angle sum rule

For any regular polygon, the sum of exterior angles is $360^\circ$. A square has 4 sides.
$\text{Single exterior angle} = \frac{360^\circ}{4}$

Step2: Calculate the value

$\frac{360^\circ}{4} = 90^\circ$

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Step1: Identify number of sides

The polygon has 7 sides.

Step2: Compute single exterior angle

$\text{Single exterior angle} = \frac{360^\circ}{7} \approx 51.4^\circ$

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Step1: Identify number of sides

The polygon has 5 sides.

Step2: Compute single exterior angle

$\text{Single exterior angle} = \frac{360^\circ}{5} = 72^\circ$

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Step1: Rearrange exterior angle formula

$n = \frac{360^\circ}{\text{Single exterior angle}}$, where $n$ is number of sides.

Step2: Substitute and calculate

$n = \frac{360^\circ}{40^\circ} = 9$

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Step1: Rearrange exterior angle formula

$n = \frac{360^\circ}{\text{Single exterior angle}}$

Step2: Substitute and calculate

$n = \frac{360^\circ}{4^\circ} = 90$

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Step1: Sum exterior angles to $360^\circ$

$x + 2x + 4x + 5x + 8x + 10x = 360$

Step2: Combine like terms

$30x = 360$

Step3: Solve for $x$

$x = \frac{360}{30} = 12$

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Step1: Sum exterior angles to $360^\circ$

$5x + 9x + 10x = 360$

Step2: Combine like terms

$24x = 360$

Step3: Solve for $x$

$x = \frac{360}{24} = 15$

Step4: Find smallest exterior angle

Smallest angle is $5x = 5 \times 15^\circ = 75^\circ$

Answer:

1. $90^\circ$
2. $51.4^\circ$
3. $72^\circ$
4. 9 sides
5. 90 sides
6. $x=12$
7. $75^\circ$