Question

The sum of the measures of the angles of a triangle is $180$. The sum of the measures of the second and third angles is twice the measure if the first angle. The third angle is twelve more than the second. Find the measures of the three angles.

Short answer

The required measures angles are$(60,54,66)$.

Step-by-step solution

Step 1. Given Information.

Let $x,y,z$be the measure of first, second, third angles.

The sum of the measures of the angles of a triangle is $180$.

$x+y+z=180$

The sum of the measures of the second and third angles is twice the measure if the first angle.

$y+z=2x$

$-2x+y+z=0$

The third angle is twelve more than the second.

$z=12+y$

$-y+z=12$

Step 2. Make a system of equations.

Consider the system of equations.

$\left\{\begin{array}{l}x+y+z=180\\ -2x+y+z=0\\ -y+z=12\end{array}\right.$

Step 3. Using Elimination method to solve system of equations.

Multiply $2$in first equation to get opposite coefficients of $x$

$2(x+y+z)=2\left(180\right)$

$2x+2y+2z=360$

Add the equations : $\left\{\begin{array}{l}2x+2y+2z=360\\ -2x+y+z=0\end{array}\right.$

$3y+3z=360$

$y+z=120$

Step 4. Add the equations.

$\left\{\begin{array}{l}y+z=120\\ -y+z=12\end{array}\right.$

Add -

$2z=132$

$z=66$

Now, substitute $z=66$in one of the equations.

$-y+z=12$

$-y+66=12$

$y=54$

Step 5. Substitute all the values to get value of x

Substitute $y,z=54,66$in the equation :

$x+y+z=180$

$x+54+66=180$

$x=180-120$

$x=60$

Solution of the measures three angles is$(60,54,66).$

Step 6. Substitute all the values in original equations to check.

Substitute $x,y,z=60,54,66$in the equations :

$x+y+z=180$

$60+54+66=180$

$180=180$

And, $-2x+y+z=0$

$-2\left(60\right)+54+66=0$

$0=0$

And, $-y+z=12$

$-54+66=12$

$12=12$

All are True.