To find the measures of the angles in the triangle, we start by using the fact that the sum of the angles in any triangle is \(180\) degrees. We have three angles in the triangle: the top angle, the bottom left angle, and the bottom right angle.
Let’s denote the angles as follows:
- Top angle = \(96^\circ\)
- Bottom left angle = \(2x^\circ\)
- Bottom right angle = \(x + 12^\circ\)
We can set up the equation based on the sum of the angles:
\[ 96 + 2x + (x + 12) = 180 \]
Now, we will simplify the equation:
\[ 96 + 2x + x + 12 = 180 \] \[ 96 + 12 = 108 \]
Thus, we can rewrite the equation as:
\[ 108 + 3x = 180 \]
Next, we will isolate \(3x\):
\[ 3x = 180 - 108 \] \[ 3x = 72 \]
Now, we will divide by \(3\) to find \(x\):
\[ x = \frac{72}{3} = 24 \]
Now that we have the value of \(x\), we can find the measures of the angles:
- Bottom left angle = \(2x = 2(24) = 48^\circ\)
- Bottom right angle = \(x + 12 = 24 + 12 = 36^\circ\)
Now we have the measures of all angles:
- Top angle: \(96^\circ\)
- Bottom left angle: \(48^\circ\)
- Bottom right angle: \(36^\circ\)
Finally, we can determine the smallest angle of the triangle, which is:
\[ \boxed{36^\circ} \]
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