Question

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

Answer 1

`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.

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

`z+86=180`

Subtract
`86` from both sides of the equation:

`z=94`

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

`x+32=94`

Subtract
`32` from both sides of the equation:

`x=62`

Let's find the value of the angle:

`(62)+32`

Add:

`94`

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

_

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

`y=94`