Question

select the correct answer from each drop-down menu.
given: △abc with altitude h
prove: \\(\frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\)
image of triangle abc with altitude h from a to bc, labeled with sides c (ab), b (ac), a (bc), and altitude h. below the image is a two-column proof table with statements and reasons:

• △abc with altitude h | given
• \\(\sin(b) = \frac{h}{c}\\), \\(\sin(c) = \frac{h}{b}\\) | definition of sine
• \\(c\sin(b) = h\\), \\(b\sin(c) = h\\) | drop-down
• drop-down | transitive property of equality
• \\(\frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\) | drop-down

Explanation:

Step1: Identify missing reason

The step $c\sin(B)=h, b\sin(C)=h$ comes from multiplying both sides of $\sin(B)=\frac{h}{c}$ and $\sin(C)=\frac{h}{b}$ by $c$ and $b$ respectively. This is the multiplication property of equality.

Step2: Identify missing statement

By transitive property, since both equal $h$, $c\sin(B)=b\sin(C)$.

Step3: Identify missing reason

Dividing both sides of $c\sin(B)=b\sin(C)$ by $bc$ gives $\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$, which uses the division property of equality.

Answer:

1. Multiplication property of equality
2. $c\sin(B) = b\sin(C)$
3. Division property of equality