To find the measure of [tex]\(\angle X\)[/tex] and its exterior angle in the triangle [tex]\(XYZ\)[/tex], let's go through the given information and utilize it step-by-step.
Given:
- [tex]\(m\angle X = (7g + 12)^\circ\)[/tex]
- The exterior angle to [tex]\(\angle X\)[/tex] measures [tex]\((2g + 60)^\circ\)[/tex]
### Step 1: Understand the relationship between an interior angle and its exterior angle
Interior and exterior angles at a point on a straight line add up to [tex]\(180^\circ\)[/tex]. Therefore, we have:
[tex]\[ (7g + 12)^\circ + (2g + 60)^\circ = 180^\circ \][/tex]
### Step 2: Formulate the equation from the angle relationships
Combine the expressions for the interior and exterior angles:
[tex]\[ 7g + 12 + 2g + 60 = 180 \][/tex]
### Step 3: Simplify the equation
Combine like terms on the left-hand side:
[tex]\[ 9g + 72 = 180 \][/tex]
### Step 4: Solve for [tex]\(g\)[/tex]
Isolate [tex]\(g\)[/tex] by subtracting 72 from both sides:
[tex]\[ 9g = 108 \][/tex]
Then, divide by 9:
[tex]\[ g = 12 \][/tex]
### Step 5: Calculate the measures of the angles
Substitute [tex]\(g = 12\)[/tex] back into the expressions for the interior and exterior angles:
- Interior angle [tex]\(m\angle X\)[/tex]:
[tex]\[ 7g + 12 = 7(12) + 12 = 84 + 12 = 96^\circ \][/tex]
- Exterior angle:
[tex]\[ 2g + 60 = 2(12) + 60 = 24 + 60 = 84^\circ \][/tex]
### Conclusion:
The measures are:
- Interior angle [tex]\(\angle X = 96^\circ\)[/tex]
- Exterior angle = [tex]\(84^\circ\)[/tex]
Thus, the correct option is:
- Interior angle [tex]\(= 96^\circ\)[/tex]; exterior angle [tex]\(= 84^\circ\)[/tex]
