Question
In kite $WEAR, \angle\text{WEA}=70^\circ\text{and}\angle\text{ARW}=80^\circ.\ $Find the remaining two angles.
✓
Answer
Given, in a kite $WEAR$, $\angle\text{WEA}=70^\circ,\angle\text{ARM}=80^\circ$
Now, by the interior angle sum property of a quadrilateral,
$\angle\text{RWE}+\angle\text{WEA}+\angle\text{EAR}+\angle\text{ARW}=360^\circ$
$\Rightarrow\angle\text{RWE}+70^\circ+\angle\text{EAR}+80^\circ=360^\circ$
$\Rightarrow\angle\text{RWE}+\angle\text{EAR}+360^\circ-150^\circ$
$\Rightarrow\angle\text{RWA}+\angle\text{RAW}+210^\circ\ \ \dots(\ \text{i})$
Now, $\angle\text{RWA}+\angle\text{RAW}\ \ \dots(\text{ii})\ [\text{RW}=\text{RA}]$
& $\angle\text{AWE}+\angle\text{WAE}\ \ \dots(\text{iii})\ [\text{WE}=\text{AE}]$
On adding eqs. $(ii)$ and
$(iii)$, we get $\angle\text{RWA}+\angle\text{AWE}+\angle\text{RAW}+\angle\text{WAE}$
$\Rightarrow\angle\text{RWE}=\angle\text{RAE}$
From eq.$(i),$
$2\angle\text{RWE}=120^\circ$
$\Rightarrow\angle\text{RWE}=150^\circ$
$\Rightarrow\angle\text{RWE}=\angle\text{RAE}=150^\circ.$
Now, by the interior angle sum property of a quadrilateral,
$\angle\text{RWE}+\angle\text{WEA}+\angle\text{EAR}+\angle\text{ARW}=360^\circ$
$\Rightarrow\angle\text{RWE}+70^\circ+\angle\text{EAR}+80^\circ=360^\circ$
$\Rightarrow\angle\text{RWE}+\angle\text{EAR}+360^\circ-150^\circ$
$\Rightarrow\angle\text{RWA}+\angle\text{RAW}+210^\circ\ \ \dots(\ \text{i})$
Now, $\angle\text{RWA}+\angle\text{RAW}\ \ \dots(\text{ii})\ [\text{RW}=\text{RA}]$
& $\angle\text{AWE}+\angle\text{WAE}\ \ \dots(\text{iii})\ [\text{WE}=\text{AE}]$
On adding eqs. $(ii)$ and
$(iii)$, we get $\angle\text{RWA}+\angle\text{AWE}+\angle\text{RAW}+\angle\text{WAE}$
$\Rightarrow\angle\text{RWE}=\angle\text{RAE}$
From eq.$(i),$
$2\angle\text{RWE}=120^\circ$
$\Rightarrow\angle\text{RWE}=150^\circ$
$\Rightarrow\angle\text{RWE}=\angle\text{RAE}=150^\circ.$
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