Question

The angles of a triangle are x, y, and \[{40^o}\]. The difference between the two angles x and y is \[{30^o}\]. Find x and y.
\[
  A.\;\;\;\;\;55^\circ ,85^\circ \\
  B.\;\;\;\;\;22^\circ ,67^\circ \\
  C.\;\;\;\;\;87^\circ ,29^\circ \\
  D.\;\;\;\;\;66^\circ ,76^\circ \\
  \]

Answer 1

Hint: As we have,\[{\text{Sum of all angles of a triangle = 18}}{{\text{0}}^{\text{o}}}\] using this approach, we will substitute the values of angles of a triangle to get an equation in x and y and we have an equation given as the difference of x and y is given, then we solve them by substitution method to get the required result.

Complete step by step answer:

It is given that, $|x - y| = 30$
Let us assume that $x > y$
Therefore,
$
  x - y = {30^o} \\
  x = {30^o} + y......(i) \\
  $
Since we know that sum of all the angles of a triangle is 180°, we can as
$x + y + {40^o} = {180^o}.......(ii)$
From equation (i) and (ii), we get
\[
  {30^o} + y + y + {40^o} = {180^o} \\
   \Rightarrow 2y = {180^o} - {40^o} - {30^o} \\
   \Rightarrow 2y = {110^o} \\
   \Rightarrow y = {55^o} \\
  \]
Substituting the value of y in equation (i), we get
$
  x = {30^o} + {55^o} \\
   \Rightarrow x = {85^o} \\
  $
Therefore, angles x and y are \[{55^o}\] and \[{85^o}\]
Hence, option (A) is correct.

Note: It was assumed that $x > y$, but it is not mentioned in the question so after getting the values of x and y, we can’t say the which one will be greater as if we assume that $x < y$
$
  y - x = {30^o} \\
  y = {30^o} + x......(iii) \\
  $
Since we know that sum of all the angles of a triangle is 180°, we can as
$x + y + {40^o} = {180^o}$
From equation (ii) and (iii), we get
\[
  {30^o} + x + x + {40^o} = {180^o} \\
   \Rightarrow 2x = {180^o} - {40^o} - {30^o} \\
   \Rightarrow 2x = {110^o} \\
   \Rightarrow x = {55^o} \\
  \]
Substituting the value of x in equation (i), we get
$
  y = {30^o} + {55^o} \\
   \Rightarrow y = {85^o} \\
  $
Therefore, angles x and y are \[{55^o}\] and \[{85^o}\]
Additional information: When it is provided that the difference between any two quantity, we cannot just say which one will be greater or lesser, so we use the expression of modulus\[\left( {|x|} \right).\]