Question

If cos(w°) = sin(z°), which of the following statements is true?

triangle ABD and ECD in which angle D measures 90 degrees, angle A equals w degrees, angle B equals x degrees, angle E equals y degrees, and angle C equals z degrees

z = x and ΔABD ≅ ΔECD
w = z and ΔABD ≅ ΔECD
z = x and ΔABD ~ ΔECD
w = z and ΔABD ~ ΔECD

Answer 1

Given that \(\cos(w^\circ) = \sin(z^\circ)\), we can use the complementary angle identity for sine and cosine:

\[
\sin(z^\circ) = \cos(90^\circ - z^\circ)
\]

Thus, we can say:

\[
\cos(w^\circ) = \cos(90^\circ - z^\circ)
\]

This implies that:

1. \(w^\circ = 90^\circ - z^\circ\) (if both angles are in the same quadrant)
2. or \(w^\circ = 90^\circ + z^\circ\) (if considering the periodic nature of cosine)

Since we are considering angles in triangles, we will stick with the first case:

\[
w + z = 90^\circ
\]

This means that:

\[
z = 90^\circ - w
\]

In a triangle, if \( \angle A = w^\circ \) and \(\angle C = z^\circ\), then \(\angle A + \angle C = 90^\circ\). Therefore, \(\angle A\) and \(\angle C\) are complementary angles.

Now considering the triangles \( \Delta ABD \) and \( \Delta ECD \):
- \( \angle D = 90^\circ \)
- \( \angle A = w^\circ \)
- \( \angle E = 90^\circ - w^\circ = z^\circ \)
- And since both angles in respective triangles are summing to 90 when added to angle D (for triangles ABD and ECD), we can infer similarity.

Thus, we can conclude:

**Statement:** \( w = z \) and \( \Delta ABD \sim \Delta ECD \) are more proper with the first conditions.

The options can be interpreted again in terms of relations:

- The similarity of the triangles implies \( \Delta ABD \sim \Delta ECD \).

The correct statement is:

**Answer:** \( w = z \) and \( \Delta ABD \sim \Delta ECD \)