Question

The sum of the acute angles of an obtuse triangle is \[{70^ \circ }\] and their difference is \[{10^ \circ }\]. The largest angle is?
1) \[{110^ \circ }\]
2) \[{105^ \circ }\]
3) \[{100^ \circ }\]
4) \[{95^ \circ }\]

Answer 1

Hint: This is a pretty straight forward problem. It can be split into two parts. First, we will solve a set of simultaneous equations to find both acute angles of the triangle. Then we will apply the theorem that the sum of all the angles in a triangle is always \[{180^ \circ }\]. This will help us find the largest obtuse angle.

Complete step by step solution:
Let the two angles(acute) be denoted as ‘a’ and ‘b’. According to the first part of the question, their sum is \[{70^ \circ }\], so we arrive at the following equation.
\[a + b = {70^ \circ }\]…………..(1)
But we also know that the difference between them is \[{10^ \circ }\], which leads us to our second equation:
\[a - b = {10^ \circ }\]…………(2)
Because we have 2 linear equations in 2 variables, we can safely use simultaneous equations to arrive at ‘a’ and ‘b’.
\[ + \]\[\begin{gathered}
 a + b = {70^ \circ } \\
  \dfrac{{(a - b = {{10}^ \circ })}}{{2a = {{80}^ \circ }}} \\
\end{gathered} \]
‘b’ cancels out in the LHS and leaves only ‘a’ to be determined.
\[\boxed{a = \dfrac{{{{80}^ \circ }}}{2} = {{40}^ \circ }}\]
Putting this value of ‘a’ back into equation (1), we can determine the value of ‘b’
Equation (1): \[a + b = {70^ \circ }\], but \[a = {40^ \circ }\], so it reduces to:
\[{40^ \circ } + b = {70^ \circ }\]
\[Therefore, \boxed{b = {{70}^ \circ } - {{40}^ \circ } = {{30}^ \circ }}\]
Now, we come to the second half of the problem let the third angle in the triangle be denoted by ‘c’
According to the theorem of sum of angles of any triangle:
\[a + b + c = {180^ \circ }\]…………………(3)
But we know ‘a and ‘b’, so substituting then in equation results in the following:
\[40 + 30 + c = {180^ \circ }\]
\[c = {180^ \circ } - {40^ \circ } - {30^ \circ }\]
\[\boxed{c = {{110}^ \circ }}\]
Since the above triangle is an obtuse angled triangle, and by comparison, we can tell that \[\boxed{c = {{110}^ \circ }}\] is the largest angle.

Hence, option A is correct.

Note: In this case, we have assumed \[a > b\] which led to the equation \[a - b = {10^ \circ }\]. However, we can also assume that \[b < a\], which will give us \[b - a = {10^ \circ }\]. This will only result in the switching of values of ‘a’ and ‘b’, i.e \[b = {40^ \circ }\] and \[a = {30^ \circ }\] in this calculation. The value of ‘c’ will remain unchanged.