Question

Which one of the following cannot be the ratio of angles in a right-angled triangle?
A) $1:2:3$
B) $1:1:2$
C) $1:3:6$
D) None of these

Answer 1

Hint:
Let x be the common angle of the right-angled triangle. Also, the sum of all angles of any triangle is $180^\circ $.
Then, observe all the given options and check for which of the options, one of the angles from the ratio of the angles given is not equal to $90^\circ $.
Thus, check all the given options and choose the correct answer.

Complete step by step solution:
Let x be the common angle of the right-angled triangle.
We know that the sum of all angles of any triangle is $180^\circ $.
Now, we will observe all the given options and check for which of the options, one of the angles from the ratio of the angles given is not equal to $90^\circ $.
In option (A), the ration given is $1:2:3$ . So, we get the angles as x, 2x and 3x.
The sum of angles must be $180^\circ $ .
 $
  \Rightarrow x + 2x + 3x = 180^\circ \\
  \Rightarrow 6x = 180^\circ \\
  \Rightarrow x = \dfrac{{180^\circ }}{6} \\
  \Rightarrow x = 30^\circ \\
 $
Thus, the given angles are $x = 30^\circ $ , $2x = 2\left( {30^\circ } \right) = 60^\circ $ and $3x = 3\left( {30^\circ } \right) = 90^\circ $ .
So, the given ratio can be the ratio of angles of a right-angled triangle as one of the angles is $90^\circ $ .
Similarly, in option (B), the ration given is $1:1:2$ . So, we get the angles as x, x and 2x.
The sum of angles must be $180^\circ $ .
 $
  \Rightarrow x + x + 2x = 180^\circ \\
  \Rightarrow 4x = 180^\circ \\
  \Rightarrow x = \dfrac{{180^\circ }}{4} \\
  \Rightarrow x = 45^\circ \\
 $
Thus, the given angles are $x = 45^\circ $ , $x = 45^\circ $ and $2x = 2\left( {45^\circ } \right) = 90^\circ $ .
So, the given ratio can be the ratio of angles of a right-angled triangle as one of the angles is $90^\circ $ .
In option (A), the ration given is $1:3:6$ . So, we get the angles as x, 3x and 6x.
The sum of angles must be $180^\circ $ .
 $
  \Rightarrow x + 3x + 6x = 180^\circ \\
  \Rightarrow 10x = 180^\circ \\
  \Rightarrow x = \dfrac{{180^\circ }}{{10}} \\
  \Rightarrow x = 18^\circ
 $
Thus, the given angles are $x = 18^\circ $ , $3x = 3\left( {18^\circ } \right) = 54^\circ $ and $6x = 6\left( {18^\circ } \right) = 96^\circ $.
So, the given ratio cannot be the ratio of angles of a right-angled triangle as none of the angles is $90^\circ $.

So, option (C) is the correct answer.

Note:
Right angled triangle:
A triangle which has one of its angles as a right angle i.e. one of the angles measures $90^\circ $, that triangle is called a right-angled triangle.

The square in corner B, suggests that it is a right angle.
Right angles triangle is one of the most used triangles in mathematics as it is used in Pythagorean theorem and sine, cosine and tangent functions.