The two walking paths form a $58^{\circ }$ interior angle at the fountain. An exterior angle formed by extending one of the paths creates a linear pair with the interior angle, so the two angles add to $180^{\circ }$. Therefore, the exterior angle is $180^{\circ }-58^{\circ }=122^{\circ }$. So the correct answer is $122^{\circ }$.

The ramp, ground, and vertical support post form a triangle. Since the support post is vertical and the ground is horizontal, the angle between them is $90^{\circ }$. The angle where the ramp meets the ground is $32^{\circ }$. The angles in a triangle add to $180^{\circ }$, so the angle between the ramp and the vertical support post is $180-90-32=58$. Therefore, the correct answer is $58$.

Adjacent angles formed by two intersecting lines are supplementary, so their measures add to $180$. Set up the equation $x+(2x+30)=180$. Combining like terms gives $3x+30=180$, so $3x=150$ and $x=50$. Vertical angles are equal, so each angle opposite the angle labeled $x^{\circ }$ also measures $50^{\circ }$. Therefore, the correct answer is $50$.

First find the slope of line $k$ using the two points:

For a line that makes an angle $\theta$ with the positive $x$-axis, the slope is $m=\tan \theta$. So

The acute angle whose tangent is 1 is $45^{\circ }$. Therefore, the correct answer is $45^{\circ }$.

When a transversal crosses two parallel lines, same-side interior angles are supplementary. That means their measures add to $180^{\circ }$. Since angle 1 is $68^{\circ }$, angle 2 is $180-68=112$. So the measure of angle 2 is $112^{\circ }$.