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{y}{x}\\).
c. list all ratios of side lengths are equal to \\(\tan(\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 $\frac{y}{x}$

For similar triangles, corresponding sides are proportional. The side $y$ corresponds to $r$, and $x$ corresponds to $q$. So the equivalent ratio is $\frac{r}{q}$.

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

The tangent of an angle in a right triangle is $\frac{\text{opposite}}{\text{adjacent}}$. For angle $\alpha$, in the first triangle: $\frac{y}{x}$; in the second triangle: $\frac{r}{q}$.

Step4: Explain trig ratio angle dependence

Since the triangles are similar, all corresponding side ratios are equal. These ratios only depend on the measure of angle $\alpha$ because similar triangles preserve the shape (angle measures) and scale side lengths proportionally, so the ratio of sides stays constant for a given angle.

Answer:

a. Both are right triangles with shared acute angle $\alpha$, so by AA similarity, they are similar.
b. $\frac{r}{q}$
c. $\frac{y}{x}$ and $\frac{r}{q}$
d. Similar triangles have proportional corresponding sides, so trigonometric ratios (side length ratios) are constant for a fixed angle $\alpha$, regardless of the triangle's overall size.