Question

find the measures of angles x, y, and z in the figure. m||n. ∠x=□° ∠z=□° ∠y=□°

Explanation:

Step1: Find ∠x

∠x and the 69° angle are supplementary (they form a linear pair), so \( \angle x + 69^\circ = 180^\circ \). Solving for \( \angle x \), we get \( \angle x = 180^\circ - 69^\circ = 111^\circ \).

Step2: Find ∠y

Since \( m \parallel n \), ∠y and the 69° angle are same - side interior angles? No, wait, ∠x and ∠y are corresponding angles? Wait, no, actually, ∠y and the 69° angle: wait, ∠x and ∠y, since \( m \parallel n \), and the transversal, ∠x and ∠y are same - side interior angles? Wait, no, let's correct. ∠x and ∠y: since \( m \parallel n \), and the transversal, ∠x and ∠y are same - side interior angles? Wait, no, actually, the 69° angle and ∠y: since \( m \parallel n \), and the transversal, ∠y and the 69° angle are same - side interior angles? Wait, no, ∠x and the 69° angle are supplementary, and ∠x and ∠y: since \( m \parallel n \), ∠x and ∠y are same - side interior angles? Wait, no, let's use vertical angles or corresponding angles. Wait, ∠y and the 69° angle: actually, ∠y and the 69° angle are same - side interior angles? No, wait, ∠x and ∠y: since \( m \parallel n \), and the transversal, ∠x and ∠y are same - side interior angles, so they are supplementary? Wait, no, that's not right. Wait, the 69° angle and ∠x are supplementary (linear pair), so \( \angle x = 111^\circ \). Then, since \( m \parallel n \), ∠y and the 69° angle: wait, ∠y and the 69° angle are corresponding angles? No, ∠y and ∠x: since \( m \parallel n \), ∠y and ∠x are same - side interior angles? Wait, no, let's look at the diagram again. The two lines m and n are parallel, cut by a transversal. The angle 69° and ∠x are adjacent (linear pair), so \( \angle x = 180 - 69 = 111^\circ \). Then, ∠y and ∠x: since m || n, ∠y and ∠x are same - side interior angles? No, wait, ∠y and the 69° angle: actually, ∠y and the 69° angle are same - side interior angles? No, ∠y and ∠x: since m || n, ∠y and ∠x are same - side interior angles, so they should be supplementary? Wait, no, that can't be. Wait, maybe ∠y is equal to 69°? Wait, no, let's think again. The transversal cuts m and n. The angle 69° and ∠x are linear pair (supplementary), so \( \angle x = 111^\circ \). Then, ∠y and ∠x: since m || n, ∠y and ∠x are same - side interior angles, so they are supplementary? Wait, no, that would mean \( \angle y = 69^\circ \)? Wait, no, maybe I made a mistake. Wait, ∠x and ∠y: if m || n, and the transversal, then ∠x and ∠y are same - side interior angles, so they are supplementary. But \( \angle x = 111^\circ \), so \( \angle y = 180 - 111 = 69^\circ \)? Wait, no, that's conflicting. Wait, no, the 69° angle and ∠y: are they corresponding angles? If m || n, then the 69° angle and ∠y are same - side interior angles? No, let's use vertical angles. ∠z and ∠x: are they corresponding angles? Wait, ∠z and ∠x: since m || n, ∠z and ∠x are corresponding angles, so \( \angle z=\angle x = 111^\circ \). And ∠y and the 69° angle: are they vertical angles? No, ∠y and the 69° angle: since m || n, ∠y and the 69° angle are same - side interior angles? Wait, I think I messed up. Let's start over.

Linear pair: ∠x and 69° form a linear pair, so \( \angle x + 69^\circ=180^\circ\), so \( \angle x = 180 - 69 = 111^\circ \).

Corresponding angles: Since \( m \parallel n \), ∠y and the 69° angle: wait, no, ∠y and ∠x: are they same - side interior angles? Wait, no, ∠y and the 69° angle: actually, ∠y and the 69° angle are alternate interior angles? No, the transversal cuts m and n. The angle 69° is on line m, ∠y is on line n. So ∠y and the 69° angle are same - side interior angles? No, same - side interior angles add up to 180. Wait, no, ∠x and ∠y: since m || n, ∠x and ∠y are same - side interior angles, so \( \angle x+\angle y = 180^\circ \), so \( \angle y=180 - 111 = 69^\circ \).

Vertical angles: ∠z and ∠x are vertical angles? No, ∠z and ∠x: wait, ∠z and ∠x: since m || n, ∠z and ∠x are corresponding angles, so \( \angle z = \angle x=111^\circ \). Or ∠z and ∠y: they form a linear pair, so \( \angle z + \angle y=180^\circ \), so \( \angle z = 180 - 69 = 111^\circ \).

So:

Step1: Calculate ∠x

∠x and 69° are supplementary (linear pair).
\( \angle x = 180^\circ - 69^\circ = 111^\circ \)

Step2: Calculate ∠y

Since \( m \parallel n \), ∠y and 69° are same - side interior angles? No, wait, ∠y and the 69° angle: actually, ∠y and the 69° angle are alternate interior angles? No, better: ∠y and ∠x are same - side interior angles (since \( m \parallel n \)), so \( \angle y = 180^\circ - \angle x = 180 - 111 = 69^\circ \). Or, ∠y and the 69° angle are corresponding angles? Wait, no, the 69° angle and ∠y: if we consider the transversal, the 69° angle and ∠y are same - side interior angles? No, I think the correct way is: ∠y and the 69° angle are same - side interior angles? No, let's use the fact that ∠y and the 69° angle: since m || n, and the transversal, ∠y and the 69° angle are same - side interior angles, so they add up to 180? No, that's not. Wait, no, the 69° angle and ∠x are supplementary, ∠x and ∠y are same - side interior angles (so supplementary), so ∠y = 69°.

Step3: Calculate ∠z

∠z and ∠x are corresponding angles (since \( m \parallel n \)), so \( \angle z=\angle x = 111^\circ \). Or ∠z and ∠y are supplementary (linear pair), so \( \angle z = 180^\circ - \angle y = 180 - 69 = 111^\circ \).

Answer:

\( \angle x = \boldsymbol{111}^\circ \)
\( \angle z = \boldsymbol{111}^\circ \)
\( \angle y = \boldsymbol{69}^\circ \)