Respuesta :
Answer:
37.9°
Step-by-step explanation:
Notice that angle Z is formed by sides XZ and YZ.
First, we need to find the slopes of each side.
[tex]m_{XZ}=\frac{2-(-1)}{1-(-1)}= \frac{2+1}{1+1}=\frac{3}{2}[/tex]
The slope of side XZ is 3/2.
[tex]m_{YZ}=\frac{2-1}{1-(-2)}=\frac{1}{1+2}=\frac{1}{3}[/tex]
The slope of side YX is 1/3.
Now, we need to recur to the angle-slope formula, which is gonna give us the angle between both sides, that is, angle Z.
[tex]tan(\angle Z)=|\frac{m_{XZ}-m_{YZ}}{1+(m_{XZ})(m_{YZ})} |[/tex]
Replacing each slope, we have
[tex]tan(\angle Z)=|\frac{\frac{3}{2}-\frac{1}{3} }{1+\frac{3}{2}(\frac{1}{3})} |\\tan(\angle Z)=|\frac{\frac{9-2}{6} }{1+\frac{1}{2} } |=|\frac{\frac{7}{6} }{\frac{3}{2} } |\\tan(\angle Z)=|\frac{14}{18} |=|\frac{7}{9} |[/tex]
Then, we solve for [tex]\angle Z[/tex]
[tex]\angle Z=tan^{-1}(\frac{7}{9} ) \approx 37.9\°[/tex]
Therefore, the approximate measure of angle Z is 37.9°
