Question

given m || n, find the value of x and y. (4x-10)° m (8x-2)° n (2y)°

Explanation:

Step1: Identify Vertical Angles or Corresponding Angles

Since \( m \parallel n \), the angle \( (4x - 10)^\circ \) and the angle \( (8x - 2)^\circ \) are same - side interior angles? Wait, no. Wait, the angle \( (4x - 10)^\circ \) and the angle adjacent to \( (8x - 2)^\circ \)? Wait, actually, the angle \( (4x - 10)^\circ \) and the angle that is vertical to the angle supplementary to \( (8x - 2)^\circ \)? Wait, no. Let's re - examine. The angle \( (4x - 10)^\circ \) and the angle that is equal to \( (4x - 10)^\circ \) (vertical angle) and \( (8x - 2)^\circ \) and \( (2y)^\circ \) are supplementary? Wait, no. Wait, \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \): since \( m\parallel n \), the angle \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \) are same - side interior angles? No, wait, actually, \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \) are same - side interior angles? Wait, no, let's see the lines. The transversal cuts \( m \) and \( n \). The angle \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \): if we consider the alternate - exterior or alternate - interior. Wait, no, actually, \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \) are same - side interior angles? Wait, no, let's calculate. Wait, the angle \( (4x - 10)^\circ \) and the angle that is vertical to it (let's call it \( A \)) and \( (8x - 2)^\circ \) and \( A \) are same - side interior angles? No, maybe \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \) are supplementary? Wait, no, that can't be. Wait, actually, \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \) are same - side interior angles? Wait, no, let's think again. The angle \( (4x - 10)^\circ \) and \( (8x - 2)^\circ \): since \( m\parallel n \), \( (4x - 10)^\circ+(8x - 2)^\circ = 180^\circ \)? Wait, no, that would be same - side interior angles. Wait, let's check: same - side interior angles are supplementary. So \( (4x - 10)+(8x - 2)=180 \)

Step2: Solve for \( x \)

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Wait, no, that gives \( x = 16 \), but let's check. If \( x = 16 \), then \( 4x-10=4\times16 - 10=64 - 10 = 54 \), \( 8x - 2=8\times16-2 = 128 - 2=126 \), and \( 54 + 126=180 \), which is correct for same - side interior angles.

Step3: Solve for \( y \)

Now, \( (8x - 2)^\circ \) and \( (2y)^\circ \) are supplementary (since they form a linear pair). We know \( x = 16 \), so \( 8x-2=126^\circ \). Then \( 126+2y = 180 \)

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Answer:

\( x = 16 \), \( y=27 \)