Sec 4 E Maths: Angles and Polygons, practice questions & worked solutions

$$\begin{aligned}\text{size of largest interior angle}&=180\degree(6-2)\times\frac8{36}\\&=160\degree\\\text{smallest exterior angle}&=180\degree-160\degree\\&=20\degree\end{aligned}$$

Answer: $20\degree$

(a)

Size of 1 int angle from octagon

$$\begin{aligned}&=\frac{(8-2)\times180^\circ}8=135^\circ\end{aligned}$$

Size of 1 int angle from Hexagon

$$\begin{aligned}&=\frac{(6-2)\times180^\circ}6=120^\circ\end{aligned}$$

$$\begin{aligned}\angle a&=360^\circ-135^\circ-120^\circ\ (\text{sum of }\angle\text{s at a pt.})\\&=105^\circ\end{aligned}$$

(b)

$$\begin{aligned}\frac{(n-2)\times180^\circ}n&=105^\circ\\n-2&=\frac{105^\circ n}{180^\circ}\\n-\frac{105^\circ n}{180^\circ}&=2\\0.41667n&=2\\n&=4.8\end{aligned}$$

Since $n$ is not a positive integer, that polygon does not exist.

Answer: (a) $a=360^\circ-135^\circ-120^\circ=105^\circ$   (b) No: $\dfrac{(n-2)\times180^\circ}{n}=105^\circ$ gives $n=4.8$, which is not a whole number

(a)

$$\begin{aligned}18x&=360^\circ\\x&=20^\circ\end{aligned}$$

(b)

$$\begin{aligned}3x+6x&=9x\\&=180^\circ\end{aligned}$$

$\therefore$ the angles are interior angles.

$$\begin{aligned}6x+5x&=11x\\&\ne180^\circ\end{aligned}$$

The quadrilateral is a trapezium and not a parallelogram as it has 2 parallel sides and 2 non-parallel sides.

Answer: (a) $x=20^\circ$ (b) a trapezium

$$\begin{aligned}\text{Interior angle of each pentagon}&=\frac{(5-2)}5\times180^\circ=108^\circ\\\text{So interior angle of polygon inside}&=360^\circ-108^\circ-108^\circ=144^\circ\\\therefore\frac{n-2}n\times180^\circ&=144^\circ\quad\text{or}\quad\frac{360^\circ}n=36^\circ\\\therefore n&=10\end{aligned}$$

Answer: $10$

(a)

$$\angle CBD=\angle CDB$$

$$=\frac{180^\circ-108^\circ}2\text{ (base $\angle$s of isosceles $\triangle$)}$$

$$=36^\circ$$

$$\text{ext. }\angle=\frac{360^\circ}n=36^\circ$$

$$n=10$$

(b)

$$\text{Interior }\angle=180^\circ-36^\circ\text{ (adj. $\angle$s on a straight line)}$$

$$=144^\circ$$

Alternative method:

$$\text{Interior }\angle=\frac{(10-2)\times180^\circ}{10}$$

$$=144^\circ$$

Answer: $\text{(a) }10\quad\text{(b) }144^\circ$

(a)

$$\begin{aligned}150^\circ+160^\circ+170^\circ+(n-3)(130^\circ)&=(n-2)(180^\circ)\\480^\circ+130n-390^\circ&=180^\circ n-360^\circ\\450^\circ&=50n\\n&=9\end{aligned}$$

(b)

Its exterior angle is $35^\circ$. $n=\dfrac{360}{35}=10\dfrac27$.

Since the number of sides is not a whole number / (positive) integer, it is not possible to be a regular polygon to have an interior angle of $145^\circ$.

Answer: (a) $n=9$   (b) each exterior angle would be $35^\circ$, and $360^\circ\div35^\circ=10\tfrac27$ is not a whole number, so no such regular polygon exists

Let interior angle of polygon be $x$.

$$3x+\angle a+\angle b=(5-2)\times180^\circ$$

$$3x+72^\circ=540^\circ$$

$$x=156^\circ$$

$$n=\frac{360^\circ}{180^\circ-156^\circ}$$

$$=15$$

Answer: $n=15$

(a)

(i)

$$\begin{aligned}\frac{(n-2)\times180}{n}&=y\\ny&=180n-360\\180n-ny&=360\\n(180-y)&=360\\n&=\frac{360}{180-y}\end{aligned}$$

(ii)  when $y=20$, $n=\dfrac94$.

Since $n$ is not an integer, it is not possible.

(b)

$$\begin{aligned}\angle ABC&=360\degree-190\degree\\&=170\degree\\\text{Ext angle}&=180\degree-170\degree\\&=10\degree\\(n-1)35\degree+10\degree&=360\degree\\35n-35+10&=360\\35n&=360+25\\n&=11\end{aligned}$$

Answer: (a)(i) $\dfrac{360}{180-y}$  (ii) Not possible  (b) $n=11$

(a)

$$\begin{aligned}\frac{180(n-2)}n\times2+\frac{180(N-2)}N&=360\\2N(n-2)+n(N-2)&=2nN\\2nN-4N+nN-2n&=2nN\\2n&=nN-4N\\N(n-4)&=2n\\N&=\frac{2n}{n-4}\end{aligned}$$

Alternative method

$$\begin{aligned}\text{Exterior }\angle\text{ of }N\text{-sided polygon}&=\frac{360}N\\\text{Interior }\angle\text{ of }n\text{-sided polygon}&=\frac{180(n-2)}n\\180+\frac{360}N&=\frac{2\times180(n-2)}n\\1+\frac2N&=\frac{2(n-2)}n\\\frac2N&=\frac{2n-4}n-1\\\frac2N&=\frac{2n-4-n}n\\\frac N2&=\frac n{n-4}\\N&=\frac{2n}{n-4}\end{aligned}$$

(b)

$$\begin{aligned}\frac{2n}{n-4}&=8\\2n&=8n-32\\6n&=32\\n&=\frac{16}3\end{aligned}$$

It is not a octagon as $n$ is not an integer.

Answer: (a) $N=\dfrac{2n}{n-4}$ (shown)   (b) $\dfrac{2n}{n-4}=8$ gives $n=\dfrac{16}{3}$, which is not a whole number, so a regular octagon cannot be formed

(a)

$$\begin{aligned}\angle EDC=\angle BAE&=\frac{(5-2)\times180}{5}=108\degree\\\angle AEB&=\frac{180-108}{2}=36\degree\\\angle BED&=108-36=72\degree\\\angle BED+\angle EDC&=72+108=180\degree\end{aligned}$$

By the converse of interior angles, $BE$ is parallel to $CD$

(b)

$\angle EBX=180-36=144\degree$
$\angle BEC=108-36-36=36\degree$
As $\angle EBX+\angle BEC=180\degree$, by the converse/property of interior angles, $EC\parallel BX$.

$BECX$ is a rhombus as $\mathbf{BE\parallel CX}$ (or $DX$) and $\mathbf{EC\parallel BX}$ and adjacent sides $BE=EC$.

Answer: (b) rhombus: $BE\parallel CX$, $EC\parallel BX$ and $BE=EC$