Question

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

Explanation:

Step1: Identify alternate interior angles

Since \( m \parallel n \), the angles \( (4x + 14)^\circ \) and \( (3x + 19)^\circ \) are alternate interior angles, so they are equal.

$$ 4x + 14 = 3x + 19 $$

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:

$$ 4x - 3x + 14 = 3x - 3x + 19 $$

$$ x + 14 = 19 $$

Subtract 14 from both sides:

$$ x + 14 - 14 = 19 - 14 $$

$$ x = 5 $$

Step3: Find \( y \)

The angle \( y^\circ \) and \( (3x + 19)^\circ \) are supplementary (linear pair), so \( y + (3x + 19) = 180 \). Substitute \( x = 5 \):

$$ 3x + 19 = 3(5) + 19 = 15 + 19 = 34 $$

Then,

$$ y + 34 = 180 $$

Subtract 34 from both sides:

$$ y = 180 - 34 = 146 $$

Answer:

\( x = 5 \), \( y = 146 \)