Question

How do you find the measures of the angles of a triangle if the measure of one angle is twice the measure of a second angle and the third angle measures 3 times the second angle decreased by 12?

Answer 1

Hint: We first use the given relations between the angles of the triangle. We assume the second angle as a variable. Based on the variable, we find the other two angles. We use the theorem of angles of triangles adding up to ${{180}^{\circ }}$. We solve the variable to find the solution.

Complete step-by-step solution:
We need to find the measures of angles of a triangle whose relations between the angles are given.
One angle is twice the measure of a second angle and the third angle measures 3 times the second angle decreased by 12.
Let’s assume the second angle being $x$. The first angle is twice the second angle.
The measure of the first angle is $2\times x=2x$.
The third angle measures 3 times the second angle decreased by 12.
The third angle is $3\times x-12=3x-12$.
We also know that the sum of all three angles of a triangle is always equal to ${{180}^{\circ }}$.
The angles in our triangle are $x,2x,3x-12$. The addition will give ${{180}^{\circ }}$.
So, $x+2x+\left( 3x-12 \right)=180$.
Now we solve the binary operations.
$\begin{align}
  & x+2x+3x-12=180 \\
 & \Rightarrow 6x-12=180 \\
 & \Rightarrow 6x=180+12=192 \\
\end{align}$
Now, we divide both sides of the equation with 6 and get
$\begin{align}
  & \dfrac{6x}{6}=\dfrac{192}{6} \\
 & \Rightarrow x=32 \\
\end{align}$
Therefore, the second angle is ${{32}^{\circ }}$.
The other angles are $2x=2\times {{32}^{\circ }}={{64}^{\circ }}$ and $3\times {{32}^{\circ }}-12={{84}^{\circ }}$.
Therefore, the angles of the triangle are ${{32}^{\circ }},{{64}^{\circ }},{{84}^{\circ }}$.

Note: The theorem of ‘sum of all three angles of a triangle is always equal to ${{180}^{\circ }}$’ is true for any kind of triangles. The binary operations and the division can also be explained in the form GCD of the divisor and the dividend.