1. Problem statement: The user asks for an explanation of the angle of elevation and the angle of depression, with a centered observer, a horizontal eye line, and lines of sight to objects above and below. 2. Definitions: Angle of elevation is the acute angle between the horizontal line through the observer's eye and the line of sight to an object above the eye. 3. Definitions: Angle of depression is the acute angle between the horizontal line through the observer's eye and the line of sight to an object below the eye. 4. Diagram orientation and important rule: Angles are measured from the horizontal through the eye, and the horizontal line is common to both angles so corresponding alternate interior angles often make angle of depression equal to an angle of elevation measured from the object's horizontal if parallel lines are in play. 5. Formula used for right-triangle situations: For either angle $\theta$ (elevation) or $\phi$ (depression), when the eye height is $h$ and the horizontal distance to the object is $d$, the tangent relation is used. 6. Display of the core trig formula: $$\tan(\theta)=\frac{h}{d}$$ 7. Explanation of the formula: This means the tangent of the angle equals the ratio of the vertical difference (opposite) to the horizontal distance (adjacent). 8. Important rule for drawing: The horizontal eye line should be dotted and the line of sight should be a straight line from the eye to the object, with the angle drawn between the dotted horizontal and that line of sight. 9. Worked numerical example showing intermediate work and cancellation: Suppose the observer's eye is $h=10$ units above a reference and the horizontal distance to an object is $d=20$ units; then 10. Compute tangent value: $\tan(\theta)=\frac{10}{20}$ 11. Show cancellation step as required: $\frac{\cancel{10}}{\cancel{20}}=\frac{1}{2}$ 12. Conclude the angle value: $\theta=\arctan\left(\frac{1}{2}\right)$ 13. Learner-friendly interpretation: If the line of sight goes upward from the eye to the object then the angle formed above the dotted horizontal is the angle of elevation. 14. Learner-friendly interpretation: If the line of sight goes downward from the eye to the object then the angle formed below the dotted horizontal is the angle of depression. 15. Final concise answer: Angle of elevation = the acute angle above the horizontal eye line to an object above. 16. Final concise answer: Angle of depression = the acute angle below the horizontal eye line to an object below.