question_answer

The measure of all the angles of a parallelogram, if an angle adjacent to the smallest angle is \[{{24}^{o}}\] less than twice the smallest angle, is

A)  \[{{37}^{o}},\,\,{{143}^{o}},\,\,{{37}^{o}},\,\,{{143}^{o}}\]

B)        \[{{108}^{o}},\,\,{{72}^{o}},\,\,{{108}^{o}},\,\,{{72}^{o}}\]

C)        \[{{68}^{o}},\,\,{{112}^{o}},\,\,{{68}^{o}},\,\,{{112}^{o}}\]

D)         None of these

Correct Answer: C

Solution :

Let the smallest angle be \[\angle A=x,\]and other adjacent angle\[\angle B={{(2x-24)}^{o}}\] Now, sum of adjacent angles of parallelogram is \[{{180}^{o}}.\] \[\therefore \]\[\angle A+\angle B={{180}^{o}}\] \[\Rightarrow \]\[x+2x-{{24}^{o}}={{180}^{o}}\] \[\Rightarrow \]\[3x={{204}^{o}}\Rightarrow x={{68}^{o}}\] \[\therefore \]\[A=x={{68}^{o}}\] and \[\angle B={{(2x-24)}^{o}}=2\times {{68}^{o}}-{{24}^{o}}={{112}^{o}}\] Since, opposite angles of a parallelogram are equal. So, \[\angle A=\angle C={{68}^{o}},\angle B=\angle D={{112}^{o}}\]