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

Answer:

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

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.

$z$ 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.

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

$y=94$