Question

quiz 7-1: angles of polygons & parallelograms
part i: angles of polygons

1. what is the sum of the degrees of the interior angles of a 19-gon?
2. if the sum of the interior angles of a polygon is $1800^\circ$, how many sides does it have?
3. what is the measure of an interior angle of a regular nonagon?
4. what is the sum of the exterior angles of a 25-gon?
5. what is the measure of each exterior angle of a regular decagon

find the value of $x$.

1. $x = $
2. $x = $
3. $x = $
4. $x = $

part ii: parallelograms

Explanation:

Step1: Sum of interior angles (19-gon)

Use formula $(n-2)\times180^\circ$, $n=19$

$$\begin{align*} (19-2)\times180^\circ&=17\times180^\circ\\ &=3060^\circ \end{align*}$$

Step2: Find sides for 1800° sum

Rearrange formula $n=\frac{S}{180^\circ}+2$, $S=1800^\circ$

$$\begin{align*} n&=\frac{1800^\circ}{180^\circ}+2\\ &=10+2\\ &=12 \end{align*}$$

Step3: Regular nonagon interior angle

First find total sum, $n=9$, then divide by 9

$$\begin{align*} \text{Total sum}&=(9-2)\times180^\circ=1260^\circ\\ \text{Single angle}&=\frac{1260^\circ}{9}=140^\circ \end{align*}$$

Step4: Sum of exterior angles

All convex polygons have exterior angle sum $360^\circ$
No calculation needed, sum is $360^\circ$

Step5: Regular decagon exterior angle

Divide total exterior sum by 10
$\frac{360^\circ}{10}=36^\circ$

Step6: Find x in 8-sided polygon

First total interior sum, $n=8$: $(8-2)\times180^\circ=1080^\circ$. Subtract known angles.

$$\begin{align*} \text{Sum of known angles}&=117+145+133+140+125+119\\ &=779^\circ\\ x&=1080^\circ-779^\circ=301^\circ \end{align*}$$

Step7: Find x in regular 8-sided polygon

First single interior angle: $\frac{(8-2)\times180^\circ}{8}=135^\circ$. Set equal to $16x+23$.

$$\begin{align*} 16x+23&=135\\ 16x&=135-23\\ 16x&=112\\ x&=\frac{112}{16}=7 \end{align*}$$

Answer:

1. $3060^\circ$
2. 12 sides
3. $140^\circ$
4. $360^\circ$
5. $36^\circ$
6. $301$
7. (No figure provided, cannot solve)
8. $7$
9. (No figure provided, cannot solve)