Question

in triangle abc, the altitude, \\(\overline{cd}\\), is labeled \\(h\\).
move options to the table to complete the proof of the law of sines.
\\(\

$$\begin{array}{|c|c|}\\hline\\text{statements}&\\text{reasons}\\\\\\hline\\_\\_\\_\\_ = \\frac{h}{a};\\_\\_\\_\\_ = \\frac{h}{b}&\\text{definition of sine}\\\\\\hline h = \\_\\_\\_\\_; h = \\_\\_\\_\\_&\\text{multiplication property of equality}\\\\\\hline \\_\\_\\_\\_ = \\_\\_\\_\\_&\\text{substitution property}\\\\\\hline \\_\\_\\_\\_ = \\_\\_\\_\\_&\\text{division property of equality}\\\\\\hline\\end{array}$$

\\)
options (to be moved to the table):
\\(\sin a\\), \\(\sin b\\), \\(a\sin a\\), \\(a\sin b\\), \\(b\sin a\\), \\(b\sin b\\), \\(\frac{\sin a}{a}\\), \\(\frac{\sin a}{b}\\), \\(\frac{\sin b}{a}\\), \\(\frac{\sin b}{b}\\)

Explanation:

Step1: Define sine for angles

$\sin A = \frac{h}{a}$; $\sin B = \frac{h}{b}$

Step2: Isolate h via multiplication

$h = a\sin A$; $h = b\sin B$

Step3: Set h expressions equal

$a\sin A = b\sin B$

Step4: Divide to get Law of Sines

$\frac{\sin A}{a} = \frac{\sin B}{b}$

Answer:

Statements	Reasons
$h = a\sin A$; $h = b\sin B$	Multiplication property of equality
$a\sin A = b\sin B$	Substitution property
$\frac{\sin A}{a} = \frac{\sin B}{b}$	Division property of equality