Question

1. triangle abc is similar to triangle def.

select all angles that have a sine of $\frac{2}{\sqrt{13}}$.
a. $\angle a$
b. $\angle b$
c. $\angle c$
d. $\angle d$
e. $\angle e$
f. $\angle f$

Explanation:

Step1: Recall the sine formula

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Analyze \(\triangle DEF\)

In \(\triangle DEF\), \(\sin E=\frac{DF}{DE}\). Given \(DF = 2\) and \(DE=13\), \(\sin E=\frac{2}{\sqrt{13}}\) (rationalizing the denominator: \(\frac{2}{\sqrt{13}}=\frac{2\sqrt{13}}{13}\)).

Step3: Use the similarity of triangles

Since \(\triangle ABC\sim\triangle DEF\), \(\angle A=\angle D\), \(\angle B=\angle E\), \(\angle C=\angle F\).

Answer:

b. \(\angle B\), e. \(\angle E\)