Question

1. here are two right triangles.

a. explain how you know the right triangles are similar.
b. write a ratio that is equivalent to \\(\frac{z}{y}\\).
c. list all ratios of side lengths that are equal to \\(\cos(\alpha)\\)?
d. explain why the values of the trigonometric ratios only depend on the angle \\(\alpha\\) and not on the particular lengths, \\(x, y, z, q, r, s\\).

Explanation:

Step1: Prove similarity via AA criterion

Both triangles are right triangles (one right angle each), and they share the common acute angle $\alpha$. By the AA (Angle-Angle) similarity postulate, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Step2: Find equivalent ratio for $\frac{x}{y}$

For similar triangles, corresponding sides are proportional. The side $x$ corresponds to $q$, and $y$ corresponds to $r$.
$\frac{x}{y} = \frac{q}{r}$

Step3: Identify $\cos(\alpha)$ ratios

Cosine of an angle in a right triangle is $\frac{\text{adjacent}}{\text{hypotenuse}}$. For angle $\alpha$, adjacent side is $x$ (first triangle) / $q$ (second triangle), hypotenuse is $z$ (first triangle) / $s$ (second triangle).
$\cos(\alpha) = \frac{x}{z} = \frac{q}{s}$

Step4: Explain trig ratio angle dependence

Similar triangles have proportional sides, so the ratio of any two corresponding sides (which defines trigonometric ratios) is constant for the same angle $\alpha$, regardless of triangle size.

Answer:

a. Both are right triangles with shared acute angle $\alpha$, so by AA similarity, they are similar.
b. $\frac{q}{r}$
c. $\frac{x}{z}$, $\frac{q}{s}$
d. The triangles are similar, so corresponding side ratios (trigonometric ratios) are equal for the same angle $\alpha$, regardless of the triangle's specific side lengths.