Question

solving for angle measures of right triangles
identifying angles of elevation and angles of depression

use the diagram to complete the statements.

the angle of depression from point r to point s is angle dropdown
the angle of elevation from point s to point r is angle 1,2,3,4 dropdown
angle 2 is the angle of elevation from blank
angle 1 is the angle of blank dropdown

Answer:

To solve this, we recall the definitions of angle of depression and angle of elevation:

• Angle of depression: The angle formed between the horizontal line from an observer and the line of sight to an object below the horizontal.
• Angle of elevation: The angle formed between the horizontal line from an observer and the line of sight to an object above the horizontal.

1. Angle of depression from \( R \) to \( S \)

From point \( R \), the horizontal line is the dashed line (angle 2’s horizontal). The line of sight to \( S \) is \( RS \), forming angle \( 3 \) (since it’s below the horizontal from \( R \)). So the angle of depression from \( R \) to \( S \) is angle \( 3 \).

2. Angle of elevation from \( S \) to \( R \)

From point \( S \), the horizontal line is the dashed line (angle 4’s horizontal). The line of sight to \( R \) is \( SR \), forming angle \( 4 \)? Wait, no—wait, the horizontal from \( S \) is parallel to the horizontal from \( R \) (alternate interior angles). Wait, actually, the angle of elevation from \( S \) to \( R \) should be equal to the angle of depression from \( R \) to \( S \) (due to parallel lines and transversal). Wait, looking at the diagram: the horizontal at \( R \) is the middle dashed line (angle 2), horizontal at \( Q \) is top dashed, horizontal at \( S \) is bottom dashed.

Wait, let’s re-express:

• Horizontal at \( R \): middle dashed line (angle 2 is between \( QR \) and this horizontal).
• Horizontal at \( S \): bottom dashed line (angle 4 is between \( SR \) and this horizontal).
• Horizontal at \( Q \): top dashed line (angle 1 is between \( QR \) and this horizontal).

Angle of elevation from \( S \) to \( R \): from \( S \), looking up to \( R \), the angle between \( S \)’s horizontal (bottom dashed) and \( SR \) is angle \( 4 \)? No, wait—angle of elevation is between the observer’s horizontal and the line of sight to the object above. From \( S \), \( R \) is above \( S \)’s horizontal, so the angle between \( S \)’s horizontal (bottom dashed) and \( SR \) is angle \( 4 \)? Wait, no, the diagram shows angle \( 3 \) at \( R \) between middle horizontal and \( SR \), and angle \( 4 \) at \( S \) between bottom horizontal and \( SR \). Since the horizontals are parallel, angle of elevation from \( S \) to \( R \) is equal to angle of depression from \( R \) to \( S \) (alternate interior angles). So angle of elevation from \( S \) to \( R \) is angle \( 4 \)? Wait, no, the options for the second blank are 1,2,3,4. Wait, maybe I misread. Let’s check again:

Wait, the first statement: “The angle of depression from point \( R \) to point \( S \) is angle ___”. From \( R \), horizontal is middle dashed (angle 2’s side). The line of sight to \( S \) is \( RS \), so the angle between horizontal (middle dashed) and \( RS \) is angle \( 3 \) (since it’s below the horizontal). So angle \( 3 \).

Second statement: “The angle of elevation from point \( S \) to point \( R \) is angle ___”. From \( S \), horizontal is bottom dashed (angle 4’s side). The line of sight to \( R \) is \( SR \), so the angle between horizontal (bottom dashed) and \( SR \) is angle \( 4 \)? Wait, no—angle of elevation is when you look up, so from \( S \), the horizontal is bottom dashed, and \( R \) is above, so the angle between bottom dashed and \( SR \) is angle \( 4 \)? But angle \( 3 \) and angle \( 4 \) are equal (alternate interior angles). Wait, maybe the options are 1,2,3,4. Let’s see the diagram:

• Angle 1: between top dashed (Q’s horizontal) and \( QR \).
• Angle 2: between middle dashed (R’s horizontal) and \( QR \).
• Angle 3: between middle dashed (R’s horizontal) and \( SR \).
• Angle 4: between bottom dashed (S’s horizontal) and \( SR \).

3. Angle 2 is the angle of elevation from ___

Angle 2 is between middle dashed (R’s horizontal) and \( QR \). So from which point is this an angle of elevation? From \( Q \)? No, \( Q \)’s horizontal is top dashed. Wait, angle of elevation from a point to \( R \): if angle 2 is the angle of elevation, then the observer is below \( R \), looking up to \( R \) along \( QR \). Wait, \( Q \) is to the left of \( R \), same horizontal? No, \( Q \)’s horizontal is top dashed, \( R \)’s horizontal is middle dashed. Wait, maybe the horizontal at \( Q \) is parallel to \( R \)’s horizontal. So angle 2 is the angle of elevation from \( Q \) to \( R \)? No, \( Q \) is on the same horizontal line (top dashed) as its own horizontal. Wait, maybe the horizontal at \( Q \) is parallel to \( R \)’s horizontal (middle dashed). So angle 2 is the angle of elevation from \( Q \) to \( R \)? No, \( Q \) is at the same height as its horizontal. Wait, maybe the horizontal at \( Q \) is the top dashed, and \( R \) is below \( Q \)’s horizontal? No, the diagram shows \( Q \) to the left, \( R \) in the middle, \( S \) to the left-bottom? Wait, no, \( S \) is below \( R \). Wait, maybe the horizontal at \( Q \) is parallel to \( R \)’s horizontal (middle dashed) and \( S \)’s horizontal (bottom dashed). So:

• Angle of depression from \( R \) to \( S \): angle between \( R \)’s horizontal (middle dashed) and \( RS \) (downward), so angle \( 3 \).
• Angle of elevation from \( S \) to \( R \): angle between \( S \)’s horizontal (bottom dashed) and \( SR \) (upward), so angle \( 4 \)? But angle \( 3 \) and \( 4 \) are equal (alternate interior angles). Wait, the options for the second blank are 1,2,3,4. Let’s check the first blank: the angle of depression from \( R \) to \( S \) is angle \( 3 \). Then the angle of elevation from \( S \) to \( R \) is angle \( 4 \)? No, maybe I got it wrong. Wait, angle of elevation from \( S \) to \( R \): when you are at \( S \), looking up to \( R \), the angle between your horizontal (bottom dashed) and the line to \( R \) ( \( SR \)) is angle \( 4 \)? But angle \( 3 \) is at \( R \), between middle dashed and \( SR \). Since the horizontals are parallel, angle of elevation from \( S \) to \( R \) is equal to angle of depression from \( R \) to \( S \) (alternate interior angles), so angle \( 3 \) and angle \( 4 \) are equal. Wait, maybe the second blank is angle \( 4 \)? No, the options are 1,2,3,4. Let’s proceed.

4. Angle 1 is the angle of ___

Angle 1 is between \( Q \)’s horizontal (top dashed) and \( QR \). Since \( R \) is below \( Q \)’s horizontal (because \( R \)’s horizontal is middle dashed, lower than \( Q \)’s), angle 1 is the angle of depression from \( Q \) to \( R \), or angle of elevation? Wait, from \( Q \), looking down to \( R \), so angle of depression.

Let’s summarize (assuming the diagram’s structure):

1. The angle of depression from \( R \) to \( S \) is angle \( 3 \).
2. The angle of elevation from \( S \) to \( R \) is angle \( 4 \)? Wait, no—maybe the angle of elevation from \( S \) to \( R \) is angle \( 3 \)? No, angle of elevation is from the lower point. Wait, \( S \) is lower than \( R \), so from \( S \), looking up to \( R \), the angle between \( S \)’s horizontal (bottom dashed) and \( SR \) is angle \( 4 \). But angle \( 3 \) is at \( R \), between middle dashed and \( SR \). Since the horizontals are parallel, angle \( 3 \) (depression from \( R \)) equals angle \( 4 \) (elevation from \( S \)) (alternate interior angles).

For the third blank: “Angle 2 is the angle of elevation from ___”. Angle 2 is between \( R \)’s horizontal (middle dashed) and \( QR \). So from which point is this an angle of elevation? If we consider the horizontal at \( Q \) (top dashed) is parallel to \( R \)’s horizontal (middle dashed), then from \( Q \), looking down to \( R \) would be depression, but angle 2 is at \( R \), between middle dashed and \( QR \). Wait, maybe the horizontal at \( Q \) is the top dashed, and \( R \) is to the right of \( Q \), same horizontal? No, the diagram shows \( Q \) to the left, \( R \) in the middle, so \( Q \) and \( R \) are on different horizontal levels? Wait, no, the top dashed is \( Q \)’s horizontal, middle dashed is \( R \)’s horizontal, bottom dashed is \( S \)’s horizontal—all parallel. So \( Q \) is at the same height as its horizontal (top dashed), \( R \) at middle, \( S \) at bottom.

Thus:

• Angle of depression from \( R \) to \( S \): \( R \)’s horizontal (middle) to \( RS \) (down to \( S \)) → angle \( 3 \).
• Angle of elevation from \( S \) to \( R \): \( S \)’s horizontal (bottom) to \( SR \) (up to \( R \)) → angle \( 4 \).
• Angle 2: \( R \)’s horizontal (middle) to \( QR \) (up to \( Q \))? No, \( Q \) is at top dashed, which is above \( R \)’s horizontal. So from \( Q \) to \( R \), it’s depression, but angle 2 is at \( R \), so angle 2 is the angle of elevation from \( Q \) to \( R \)? Wait, no—angle of elevation is from the observer to the object above. If \( Q \) is above \( R \), then from \( R \) to \( Q \) is elevation? No, \( R \) is below \( Q \), so from \( R \), looking up to \( Q \), angle 2 is the angle of elevation from \( R \) to \( Q \). But the blank says “Angle 2 is the angle of elevation from ___”, so the observer is the blank, looking up to \( R \)? Wait, no—angle of elevation is from the observer to the object. So if angle 2 is the angle of elevation, the observer is below \( R \), looking up to \( R \) along \( QR \). But \( Q \) is above \( R \), so that doesn’t make sense. Wait, maybe the horizontal at \( Q \) is parallel to \( R \)’s horizontal, so angle 2 (at \( R \)) and angle 1 (at \( Q \)) are alternate interior angles? No, angle 1 is at \( Q \), between top dashed and \( QR \); angle 2 is at \( R \), between middle dashed and \( QR \). Since top and middle dashed are parallel, angle 1 and angle 2 are equal (alternate interior angles).

For the fourth blank: “Angle 1 is the angle of ___”. From \( Q \), looking down to \( R \), so angle of depression.

Putting it all together (assuming standard definitions):

1. The angle of depression from \( R \) to \( S \) is angle \( 3 \).
2. The angle of elevation from \( S \) to \( R \) is angle \( 4 \).
3. Angle 2 is the angle of elevation from \( Q \) to \( R \) (or \( Q \), but need to check the diagram’s labels). Wait, maybe the third blank is \( Q \), but the options are not given. Wait, the original problem has dropdowns, so let’s infer:

• Angle of depression from \( R \) to \( S \): angle \( 3 \) (between \( R \)’s horizontal and \( RS \), downward).
• Angle of elevation from \( S \) to \( R \): angle \( 4 \) (between \( S \)’s horizontal and \( SR \), upward).
• Angle 2: angle of elevation from \( Q \) to \( R \) (since \( Q \) is above \( R \)’s horizontal? No, \( Q \) is at top dashed, \( R \) at middle, so \( Q \) is above \( R \), so from \( R \) to \( Q \) is elevation, but angle 2 is at \( R \). Wait, maybe the horizontal at \( Q \) is the same as \( R \)’s horizontal? No, the diagram shows three dashed lines (top, middle, bottom) parallel.

Alternatively, maybe:

• Angle of depression from \( R \) to \( S \): angle \( 3 \).
• Angle of elevation from \( S \) to \( R \): angle \( 3 \) (since alternate interior angles, but that would mean angle \( 4 = 3 \)). Wait, no—angle of elevation from \( S \) to \( R \) is equal to angle of depression from \( R \) to \( S \) (because the horizontal lines are parallel, so the angles are congruent). So angle of elevation from \( S \) to \( R \) is angle \( 3 \)? But angle \( 3 \) is at \( R \), and angle of elevation from \( S \) is at \( S \). Wait, no—angle of elevation is measured at the observer’s position. So at \( S \), the angle between \( S \)’s horizontal (bottom dashed) and \( SR \) is angle \( 4 \), which is equal to angle \( 3 \) (alternate interior angles). So angle of elevation from \( S \) to \( R \) is angle \( 4 \), and angle of depression from \( R \) to \( S \) is angle \( 3 \).

Final Answers (based on standard definitions):

1. The angle of depression from \( R \) to \( S \) is angle \( \boldsymbol{3} \).
2. The angle of elevation from \( S \) to \( R \) is angle \( \boldsymbol{4} \).
3. Angle 2 is the angle of elevation from \( \boldsymbol{Q} \) (or the point on the top horizontal, \( Q \)) to \( R \).
4. Angle 1 is the angle of \( \boldsymbol{\text{depression}} \) (from \( Q \) to \( R \), since \( R \) is below \( Q \)’s horizontal).

(Note: The exact labels depend on the diagram’s precise layout, but the key is using the definitions of angle of elevation (upward from horizontal) and angle of depression (downward from horizontal), with parallel horizontals creating congruent alternate interior angles.)