We are given a triangle with one angle labeled as $42^\circ$. We are asked to find the value of the angle at the bottom right, which is labeled $C$. We are also given the information that $9.97$ is one of the side lengths of the triangle. Since we don't have the lengths of two sides, we can assume this is an isosceles triangle.

1. Problem Description

We are given a triangle with one angle labeled as $42^\circ$. We are asked to find the value of the angle at the bottom right, which is labeled $C$. We are also given the information that $9.97$ is one of the side lengths of the triangle. Since we don't have the lengths of two sides, we can assume this is an isosceles triangle.

2. Solution Steps

If we assume the triangle is isosceles with the two sides adjacent to angle $C$ equal, then the angle at the top would not be the vertex angle and therefore the two base angles are not equal. If we assume the given side of length $9.97$ is one of the two equal sides of the isosceles triangle, and the given $42^{\circ}$ angle is the vertex angle, the angle $C$ and the other unknown angle will be equal. The sum of the angles in a triangle is $180^{\circ}$. Let's denote the unknown angle as $x$.

We have

$42^{\circ} + x + x = 180^{\circ}$

$42^{\circ} + 2x = 180^{\circ}$

$2x = 180^{\circ} - 42^{\circ}$

$2x = 138^{\circ}$

$x = \frac{138^{\circ}}{2}$

$x = 69^{\circ}$

If we assume the $42^\circ$ is a base angle, then the other base angle is $42^\circ$, so $C = 42^\circ$.

Then, the third angle will be $180^\circ - 42^\circ - 42^\circ = 180^\circ - 84^\circ = 96^\circ$.

However, the diagram shows angle $C$ looks obtuse. Hence $C$ is $69^\circ$.

3. Final Answer

$69^\circ$