1) To find [tex]x[/tex] we are going to use the Pythagorean equation:
[tex]x= \sqrt{a^{2}+b^{2} } [/tex]
where
[tex]a[/tex] and [tex]b[/tex] are the legs of our triangle. For our picture we can infer that [tex]a=10[/tex] and [tex]b=8[/tex], so lets replace those values in our equation to find [tex]x[/tex]:
[tex]x= \sqrt{10^{2}+8^{2}} [/tex]
[tex]x= \sqrt{100+64} [/tex]
[tex]x= \sqrt{164} [/tex]
[tex]x=12.8[/tex]
We can conclude that the value of [tex]x[/tex] in our triangle is 12.8
2) To find [tex]y[/tex] we are going to use the trigonometric function tangent. Remember that [tex]tan(y)= \frac{opposite}{adjacent} [/tex]. We know that the opposite side of our angle [tex]y[/tex] is 8, and its adjacent side is 10, so lets replace those values in our tangent function to find [tex]y[/tex]:
[tex]tan(y)= \frac{8}{10} [/tex]
[tex]tan(y)=0.8[/tex]
Since we need the measure of angle [tex]y[/tex], we are going to take inverse tangent to both sides to find it:
[tex]y=arctan(0.8)[/tex]
[tex]y=38.66[/tex]
We can conclude that the value of [tex]y[/tex] in our triangle is 38.66°
3) Finally, to find [tex]z[/tex] we are going to take advantage of two facts: the sum of the interior angles of a triangle is always 180°, and our triangle is a right one, so one of its sides is 90°. Therefore, [tex]y+z+90=180[/tex]. Since we already know the value of [tex]y[/tex], lets replace it in our equation and solve for [tex]z[/tex]:
[tex]38.66+z+90=180[/tex]
[tex]z+128.66=180[/tex]
[tex]z=51.34[/tex]
We can conclude that the measure of angle [tex]z[/tex] is 51.34°
