Identify the values of variables x, y, and z based on the image given of the angles formed by a transversal and parallel lines

Question

asked Dec 1, 2021 83.6k views

1 Answer

Dew Time · Answer 1 · 2021-12-08T07:52:02+0000

Answer:

$x=62\\y=94\\z=94$

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 (=).

_

Now, let's look at the figure and apply these theorems.

and
are same-side exterior angles, meaning they're supplementary:

z+86=180

Subtract
from both sides of the equation:

z=94

Now that we know our
value, let's recognize that
(
)corresponds to
x+32 , meaning they're congruent:

x+32=94

Subtract
from both sides of the equation:

x=62

Let's find the value of the angle:

(62)+32

Add:

Since the corresponding angles are of the same value, they're congruent, proving our
&
values correct.

_

Since
and the angle represented by the expression
x+32 (
°) are alternate-interior angles, they are congruent:

y=94