Question

question given m || n, find the value of x and y.
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Explanation:

Step1: Use the property of parallel lines (corresponding angles)

Since \(m\parallel n\), the corresponding angles \((4x - 5)^{\circ}\) and \((9x-10)^{\circ}\) are equal.
So, \(4x-5=9x - 10\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(-5 = 5x-10\).
Add \(10\) to both sides: \(5 = 5x\).
Divide both sides by \(5\): \(x = 1\).

Step3: Use the property of linear - pair (supplementary angles)

The angles \((y + 20)^{\circ}\) and \((4x-5)^{\circ}\) form a linear - pair (they are supplementary).
Substitute \(x = 1\) into \((4x-5)^{\circ}\), we get \(4\times1-5=-1^{\circ}\) (This is wrong, we should use the property of alternate - interior angles and linear - pair correctly.

Correct approach:
Since \(m\parallel n\), \((9x - 10)^{\circ}\) and \((y + 20)^{\circ}\) are alternate - exterior angles (equal). And \((4x-5)^{\circ}\) and \((9x - 10)^{\circ}\) are same - side interior angles (supplementary).

For \(x\):

\((4x-5)+(9x - 10)=180\) (same - side interior angles are supplementary when two parallel lines are cut by a transversal)
\(4x-5 + 9x-10=180\)
\(13x-15 = 180\)
\(13x=180 + 15\)
\(13x=195\)
\(x = 15\)

For \(y\):

Since \(m\parallel n\), \((y + 20)=(9x - 10)\) (alternate - exterior angles)
Substitute \(x = 15\) into \((y + 20)=(9x - 10)\)
\(y+20=9\times15-10\)
\(y+20=135 - 10\)
\(y+20=125\)
\(y=105\)

Answer:

\(x = 15,y = 105\)