Question

a weather balloon is 529 meters above the ground. two weather stations, a and b, monitor the balloon’s progress. compare the measures of the different angles of elevation and depression that are labeled in the diagram. which statements are true? check all that apply.
□ ( x > y )
□ ( v > x )
□ ( w = y )
□ ( y < v )
□ ( v = w )
□ ( x < w )

Explanation:

Step1: Recall tangent function

For angle \( x \) (at station A), \( \tan x = \frac{529}{229} \). For angle \( y \) (at station B), \( \tan y = \frac{529}{347} \). Since \( \frac{529}{229} > \frac{529}{347} \), and tangent is increasing in \( (0, \frac{\pi}{2}) \), so \( x > y \).

Step2: Analyze alternate interior angles

Angle \( w \) (depression from balloon to B) and angle \( y \) (elevation from B to balloon) are alternate interior angles, so \( w = y \).

Step3: Compare \( v \) and \( x \)

Angle \( v \) (depression from balloon to A) and angle \( x \) (elevation from A to balloon) are alternate interior angles, so \( v = x \). Wait, no—wait, \( v \) and \( x \): Wait, the horizontal line is at the balloon. So angle \( v \) is equal to \( x \) (alternate interior), and angle \( w \) equal to \( y \). Then, since \( x > y \), \( v = x > y = w \). So \( v > x \) is false (since \( v = x \)), \( w = y \) is true, \( y < v \) (since \( v = x > y \)) is true, \( v = w \) is false (since \( v = x > y = w \)), \( x < w \) is false (since \( x > w \)). Also, \( x > y \) is true.

Wait, let's re-express:

• \( \tan x = \frac{529}{229} \approx 2.31 \), \( \tan y = \frac{529}{347} \approx 1.52 \). So \( x > y \) (true).

• \( w \) and \( y \): alternate interior angles (horizontal line, transversal), so \( w = y \) (true).

• \( y < v \): since \( v = x \) (alternate interior), and \( x > y \), so \( y < v \) (true).

• \( v > x \): false, \( v = x \).

• \( v = w \): false, \( v = x > y = w \).

• \( x < w \): false, \( x > w \).

So true statements: \( x > y \), \( w = y \), \( y < v \).

Answer:

The true statements are: \( x > y \), \( w = y \), \( y < v \) (corresponding checkboxes: \( \boldsymbol{x > y} \), \( \boldsymbol{w = y} \), \( \boldsymbol{y < v} \))