Answer:
[tex]x=62\\y=94\\z=94[/tex]
Step-by-step explanation:
Let's start by identifying relevant angle theorems for two parallel lines cut by a transversal.
Alternate Interior Angles Theorem
Two angles that are on different sides of the transversal and inside of the two parallel lines are congruent (=).
Alternate Exterior Angles Theorem
Two angles that are on different sides of the transversal and outside of the two parallel lines are congruent (=).
Same-side Interior Angles Theorem
Two angles that are on the same side of the transversal and inside of the two parallel lines are supplementary (= 180).
Same-side Exterior Angles Theorem
Two angles that are on the same side of the transversal and outside of the two parallel lines are supplementary (=180).
Corresponding Angles Theorem
Two angles on a figure that correspond are congruent (=).
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Now, let's look at the figure and apply these theorems.
[tex]z[/tex] and [tex]86[/tex] are same-side exterior angles, meaning they're supplementary:
[tex]z+86=180[/tex]
Subtract [tex]86[/tex] from both sides of the equation:
[tex]z=94[/tex]
Now that we know our [tex]z[/tex] value, let's recognize that [tex]z[/tex] ([tex]94[/tex])corresponds to [tex]x+32[/tex], meaning they're congruent:
[tex]x+32=94[/tex]
Subtract [tex]32[/tex] from both sides of the equation:
[tex]x=62[/tex]
Let's find the value of the angle:
[tex](62)+32[/tex]
Add:
[tex]94[/tex]
Since the corresponding angles are of the same value, they're congruent, proving our [tex]x[/tex] & [tex]z[/tex] values correct.
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Since [tex]y[/tex] and the angle represented by the expression [tex]x+32[/tex] ([tex]94[/tex]°) are alternate-interior angles, they are congruent:
[tex]y=94[/tex]
