Two triangles DEF and GHK are such that $\angle\text{D}=48^\circ$ and $\angle\text{H}=57^\circ.$ If $\triangle\text{DEF}\sim\triangle\text{GHK}$ then find the measure of $\angle\text{F}.$
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Given that $\triangle\text{DEF}\sim\triangle\text{GHK}.$
$\angle\text{D}=\angle\text{G}=48^\circ\dots(\text{Given})$
$\angle\text{E}=\angle\text{H}=57^\circ\dots(\text{Given})$
In $\triangle\text{DEF},$
$\angle\text{D}+\angle\text{E}+\angle\text{F}=180^\circ\dots(\text{Angel Sum Property})$
$\Rightarrow48^\circ+57^\circ+\angle\text{F}=180^\circ$
$\Rightarrow\angle\text{F}=75^\circ$
$\angle\text{D}=\angle\text{G}=48^\circ\dots(\text{Given})$
$\angle\text{E}=\angle\text{H}=57^\circ\dots(\text{Given})$
In $\triangle\text{DEF},$
$\angle\text{D}+\angle\text{E}+\angle\text{F}=180^\circ\dots(\text{Angel Sum Property})$
$\Rightarrow48^\circ+57^\circ+\angle\text{F}=180^\circ$
$\Rightarrow\angle\text{F}=75^\circ$
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