Question

given ( m parallel n ), find the value of ( x ) and ( y ).

Explanation:

Step1: Find x using supplementary angles

x and 49° are supplementary (form a linear pair), so \( x + 49 = 180 \).
Solving for x: \( x = 180 - 49 = 131 \).

Step2: Find y using corresponding angles (since m || n)

y and x are same - side interior angles? Wait, no. Wait, actually, the angle equal to 49° and y: since m || n, the angle adjacent to y (vertical angle with the angle at n) and y: wait, better: x and y, since m || n, the angle with measure x and y: wait, no, the angle that is 49° and y: since m || n, the alternate interior angles or corresponding angles. Wait, actually, x and y: since the transversal cuts m and n (parallel), the angle x and y: wait, no, the angle adjacent to 49° (the vertical angle of 49°) and y: since m || n, y is equal to the angle that is supplementary to 49°? Wait, no, let's re - examine.

Wait, the angle marked 49° and the angle x are supplementary (linear pair), so x = 180 - 49 = 131. Then, since m || n, the angle y and the angle that is equal to 49° (vertical angle or corresponding) - wait, no, the angle y and the angle with measure 49°: since m || n, the alternate interior angles: the angle y and the angle that is 49°? No, wait, the angle x and y: since m || n, and the transversal, the same - side interior angles? Wait, no, actually, the angle y and the angle that is 49°: let's look at the diagram. The line m and n are parallel, cut by a transversal. The angle at m (y) and the angle at n (the angle adjacent to 49°) are corresponding angles. Wait, the angle adjacent to 49° is x (131°), no. Wait, I made a mistake. Let's start over.

The angle of 49° and x are supplementary (linear pair), so x = 180 - 49 = 131°. Now, since m || n, the angle y and the angle of 49°: are they corresponding angles? Wait, no, the angle y and the angle that is equal to 49° (vertical angle) - no, the angle y and x: since m || n, and the transversal, the same - side interior angles? Wait, no, the angle y and the angle with measure 49°: let's see, the angle y and the angle of 49°: if we consider the transversal, the angle y and the angle of 49° are alternate interior angles? No, alternate interior angles are equal. Wait, no, the angle y and x: since m || n, and the transversal, the same - side interior angles should be supplementary? Wait, no, x is 131°, and if we look at the diagram, y and the angle that is 49°: actually, y is equal to 131°? No, wait, no. Wait, the angle y and the angle x: since m || n, and the transversal, the angle y and the angle x: are they corresponding angles? Wait, no, the angle x is 131°, and y: let's use the fact that vertical angles are equal and corresponding angles are equal.

The angle adjacent to 49° (vertical angle) is 49°, and since m || n, the angle y and the angle of 49°: no, wait, the angle y and the angle x: since m || n, and the transversal, the angle y and the angle x are same - side interior angles? No, same - side interior angles are supplementary. Wait, x is 131°, so if y and x are same - side interior angles, then y + x = 180? But that would mean y = 49°, which contradicts. Wait, I see my mistake. The angle x and the angle that is vertical to y: no, let's look at the vertical angles. The angle y and the angle that is equal to x? No, I think I messed up the first step.

Wait, the angle of 49° and x are adjacent and form a linear pair, so x + 49 = 180, so x = 131. Then, since m || n, the angle y and the angle of 49°: are they alternate interior angles? Wait, no, the angle y and the angle that is 49°: if we look at the diagram, the line m and n are parallel, the transversal cuts them. The angle at m (y) and the angle at n (the angle opposite to 49°) - no, the angle y is equal to the angle that is 49°? No, that can't be. Wait, no, the angle y and x: since m || n, and the transversal, the angle y and x are corresponding angles? No, corresponding angles are equal. Wait, x is 131°, so if y is equal to x, then y = 131°, but that would mean same - side interior angles are equal, which is only true if they are 90°, so that's wrong.

Wait, I think I made a mistake in identifying the angles. Let's correct: The angle marked 49° and the angle y: since m || n, they are alternate interior angles? No, alternate interior angles are equal. Wait, no, the angle 49° and y: let's look at the transversal. The line m and n are parallel. The angle at n (49°) and the angle at m (y) - if the transversal is crossing them, then y and 49° are corresponding angles? No, corresponding angles are equal. Wait, no, the angle 49° and the angle y: if we consider the vertical angle of 49°, which is also 49°, and then since m || n, the angle y and the vertical angle of 49°: are they same - side interior angles? Same - side interior angles are supplementary. So y + 49 = 180? No, that would be same - side interior angles. Wait, same - side interior angles: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So the angle 49° and y are same - side interior angles? Then y = 180 - 49 = 131? But that's the same as x. Wait, no, the angle x and y: x is 131, y is 49? No, I'm getting confused.

Wait, let's start over with the correct angle relationships.

1. Find x:
The angle of 49° and x are adjacent and form a linear pair (they are on a straight line), so they are supplementary. So \( x + 49^{\circ}=180^{\circ} \)
\( x = 180 - 49=131^{\circ} \)

2. Find y:
Since \( m\parallel n \), the angle y and the angle of 49° are alternate interior angles? No, alternate interior angles are equal. Wait, no, the angle y and the angle x: since \( m\parallel n \), and the transversal, the angle y and the angle that is equal to 49° (vertical angle) - no, the angle y and the angle 49°: let's look at the diagram again. The line m and n are parallel, cut by a transversal. The angle at m (y) and the angle at n (49°) - if the transversal is going from bottom left to top right, then the angle 49° and y are corresponding angles? No, corresponding angles are equal. Wait, I think the mistake is in the initial assumption. The angle x and y: since \( m\parallel n \), and the transversal, the angle y and the angle that is equal to x? No, let's use vertical angles. The angle opposite to y (vertical angle) and x: since \( m\parallel n \), they are same - side interior angles? No, I think the correct approach is:

The angle x and the angle y: since \( m\parallel n \), and the transversal, the angle y is equal to the angle that is 49°? No, that can't be. Wait, no, the angle 49° and y: are they alternate interior angles? If the transversal is crossing m and n, then the angle 49° (at n) and y (at m) are alternate interior angles, so y = 49°? But that contradicts the earlier step. Wait, I see the problem. I misidentified the angles.

Wait, the angle marked 49° and the angle x: are they vertical angles? No, they are adjacent, forming a linear pair. So x + 49 = 180, so x = 131. Then, the angle y and the angle 49°: since m || n, they are alternate interior angles, so y = 49°? But that would mean x and y are supplementary? 131+49 = 180, which makes sense because they are same - side interior angles. Ah! Yes, same - side interior angles are supplementary. So when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So x and y are same - side interior angles, so x + y = 180. But we found x = 131, so y = 180 - 131 = 49? Wait, no, that's the opposite. Wait, no, let's draw this mentally. The two parallel lines m and n, cut by a transversal. The angle at n: 49° and x (131°) are supplementary (linear pair). The angle at m: y. Since m || n, the angle y and the angle 49° are alternate interior angles (equal), and the angle y and x are same - side interior angles (supplementary). So y = 49°, x = 131°.

Yes, that makes sense. So step 1: x and 49° are supplementary (linear pair), so x = 180 - 49 = 131. Step 2: since m || n, y and 49° are alternate interior angles, so y = 49°.

Answer:

x = 131, y = 49