Question

a weather balloon is 529 meters above the ground. two weather stations, a and b, monitor the balloons 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.
$square$ $x > y$
$square$ $v > x$
$square$ $w = y$
$square$ $y < v$
$square$ $v = w$
$square$ $x < w$
the horizontal distance from station a to the point directly below the balloon is 229 m, and from that point to station b is 347 m.

Explanation:

Step1: Relate angles via trigonometry

For right triangles, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.
For angle $x$: $\tan(x)=\frac{529}{229}$
For angle $y$: $\tan(y)=\frac{529}{347}$

Step2: Compare $\tan(x)$ and $\tan(y)$

Since $\frac{529}{229} > \frac{529}{347}$, $\tan(x)>\tan(y)$. As $\tan$ increases for $0^\circ<\theta<90^\circ$, $x>y$.

Step3: Use alternate interior angles

The dashed horizontal line is parallel to ground $AB$. So $v=x$, $w=y$.

Step4: Evaluate each statement

1. $x>y$: True (from Step2)
2. $v>x$: False ($v=x$)
3. $w=y$: True (alternate interior angles)
4. $yy$ so $y
5. $v=w$: $v=x$, $w=y$, $x

eq y$: False

1. $xy$: False

Answer:

A. $x > y$
C. $w = y$
D. $y < v$