Because the camera rotates at a constant rate, its actual direction changes by equal amounts over equal time intervals. The times increase by $3$ seconds each: from $2$ to $5$, from $5$ to $8$, and from $8$ to $11$.

The recorded angle is the smaller angle between the camera and the straight wall, so each recorded value must be between $0^{\circ }$ and $90^{\circ }$. As the camera keeps rotating, that smaller angle increases up to $90^{\circ }$ and then decreases, repeating in a predictable way.

From $2$ to $5$ seconds, the recorded angle changes from $26^{\circ }$ to $65^{\circ }$, an increase of $39^{\circ }$. Since the turning rate is constant, in the next $3$ seconds the actual direction changes by the same amount.

At $5$ seconds, the camera is at $65^{\circ }$ to the wall. Adding $39^{\circ }$ gives an actual direction of $104^{\circ }$ relative to the wall's direction. But the sensor records the smaller angle with the wall, so it records

Thus, $x=76$.

Check with the last value: from $8$ to $11$ seconds, another $39^{\circ }$ of rotation takes the actual direction from $104^{\circ }$ to $143^{\circ }$. The smaller angle with the wall is

which matches the table. Therefore, the value of $x$ is $76$.

Because $\ell \parallel m$, the acute angle at $P$ and the obtuse interior angle at $Q$ on the same side of the transversal are supplementary. So

which gives

Now substitute into the acute angle at $P$:

When a transversal cuts parallel lines, all acute angles formed are congruent. Therefore, the acute angle at $Q$ is also $58^{\circ }$. So the statement that must be true is $x=22$, and the acute angle at $Q$ measures $58^{\circ }$.

When two straight lines intersect, adjacent angles form a linear pair, so their measures add to $180^{\circ }$. If one angle is $68^{\circ }$, then the adjacent angle is $180^{\circ }-68^{\circ }=112^{\circ }$. Therefore, the correct answer is $112^{\circ }$.

The angles in any triangle add to $180^{\circ }$. Since one angle is a right angle, it measures $90^{\circ }$. Another angle measures $x^{\circ }$. So the third angle is $180-90-x=90-x$. Therefore, the correct expression is $90-x$.

Because $r\parallel s$, same-side interior angles formed by transversal $t$ are supplementary. The acute angle at line $r$ is $3x+12$, so the same-side interior angle at line $s$ must satisfy

Simplifying gives

Now evaluate the two given angles:

Since the first given angle was stated to be acute, the acute angles are $78^{\circ }$ and the obtuse angles are $102^{\circ }$. Therefore, the statement that must be true is that the obtuse angles formed by line $t$ are each $102^{\circ }$.