To find the measures of the three angles of the triangle, let's translate the problem into equations and solve them step-by-step. Let [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] represent the measures of the first, second, and third angles, respectively.
Here's what we know:
1. The sum of the measures of the angles of the triangle is 180 degrees:
[tex]\[ x + y + z = 180 \][/tex]
2. The sum of the measures of the second and third angles is two times the measure of the first angle:
[tex]\[ y + z = 2x \][/tex]
3. The third angle is 12 degrees more than the second angle:
[tex]\[ z = y + 12 \][/tex]
Now, we will use these three equations to solve for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
First, let's substitute the third equation [tex]\( z = y + 12 \)[/tex] into the second equation [tex]\( y + z = 2x \)[/tex]:
[tex]\[ y + (y + 12) = 2x \][/tex]
[tex]\[ 2y + 12 = 2x \][/tex]
Divide both sides by 2:
[tex]\[ y + 6 = x \][/tex]
So, we have:
[tex]\[ x = y + 6 \][/tex]
Next, substitute [tex]\( x = y + 6 \)[/tex] and [tex]\( z = y + 12 \)[/tex] into the first equation [tex]\( x + y + z = 180 \)[/tex]:
[tex]\[ (y + 6) + y + (y + 12) = 180 \][/tex]
Combine like terms:
[tex]\[ 3y + 18 = 180 \][/tex]
Subtract 18 from both sides:
[tex]\[ 3y = 162 \][/tex]
Divide both sides by 3:
[tex]\[ y = 54 \][/tex]
Now that we have [tex]\( y \)[/tex], we can find [tex]\( x \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ x = y + 6 = 54 + 6 = 60 \][/tex]
[tex]\[ z = y + 12 = 54 + 12 = 66 \][/tex]
Therefore, the measures of the three angles are:
[tex]\[ x = 60 \][/tex]
[tex]\[ y = 54 \][/tex]
[tex]\[ z = 66 \][/tex]
