Question

One of the angles of the triangle is \[40^\circ \]. The difference of the other two angles is \[20^\circ \]. Find the angles.

Answer 1

Hint:We know that the sum of all three angles of a triangle is \[{\text{180}}^\circ \]. Here, we are given one of the angles so if we assume two variables for the other two angles, we can have a linear equation by using the sum of three angles of a triangle.

Moreover, the difference of the other two angles is given which will give us one more linear equation. Thus, we will have two linear equations with two variables which can be easily solved to get the required answer.

Complete step by step answer:
Here, one of the angles of a triangle is given which is\[40^\circ \].
Let the other two angles be \[x^\circ \]and \[y^\circ \].
As we know that the sum of all three angles of a triangle is \[{\text{180}}^\circ \], we can say that
$
40 + x + y = 180 \\
\Rightarrow x + y = 180 - 40 \\
\Rightarrow x + y = 140 \\
$
Now, the difference of the other two angles is \[20^\circ \]. Therefore, we can say that
$x - y = 20$

Now we have two linear equations $x + y = 140$ and $x - y = 20$.
We will now solve these equations to find the values of $x$ and $y$.
From the second equation $x - y = 20$, we can say that $x = 20 + y$

Putting this value in the first equation $x + y = 140$, we get
$
20 + y + y = 140 \\
\Rightarrow 2y = 140 - 20 \\
\Rightarrow 2y = 120 \\
\Rightarrow y = 60 \\

$
We know that
$x = 20 + y$
Putting $y = 60$
$
\Rightarrow x = 20 + 60 \\
\Rightarrow x = 80 \\
$

Thus, the other two angles of the triangle are \[80^\circ \]and \[60^\circ \].

Note: In this type of question, we can check our answer by putting the obtained values in the given conditions.

In this particular question, we can check our answer by two methods. First, if we add all the angles of the triangle, it should be \[180^\circ \]. Here, we have \[40^\circ + 80^\circ + 60^\circ = 180^\circ \].

Thus our first condition is satisfied. Now, the difference of the obtained angles should be \[20^\circ \]. We have \[80^\circ - 60^\circ = 20^\circ \]. Thus, both the conditions satisfy and hence our answer is correct.