Question

1. if ( l parallel m ), find the value of each missing variable(s).

Explanation:

Step1: Use the property of parallel lines

When \(l\parallel m\), the corresponding angles are equal. So \(9x + 25=13x-19\).

$$\begin{align*} 9x+25&=13x - 19\\ 25 + 19&=13x-9x\\ 44&=4x\\ x& = 11 \end{align*}$$

Step2: Find the value of \(y\)

Also, since \(l\parallel m\), the vertical - angle relationship. Assume the angle \((13x - 19)^{\circ}\) and \((17y + 5)^{\circ}\) are supplementary (if they are same - side interior angles) or equal (if they are alternate interior or corresponding angles). Let's assume they are equal (for the sake of the problem, if we consider the general case of parallel lines and transversal). Substitute \(x = 11\) into \(13x-19\), we get \(13\times11-19=143 - 19=124\). Then \(17y+5 = 124\)

$$\begin{align*} 17y+5&=124\\ 17y&=124 - 5\\ 17y&=119\\ y&=7 \end{align*}$$

Wait, there is a mistake. Let's re - check. If \(9x + 25\) and \(13x-19\) are same - side interior angles, then \(9x + 25+13x-19 = 180\) (since same - side interior angles are supplementary when lines are parallel)

$$\begin{align*} 9x+25 + 13x-19&=180\\ 22x+6&=180\\ 22x&=174\\ x&=\frac{174}{22}=\frac{87}{11}\approx7.91 \end{align*}$$

This is wrong. Let's assume \(9x + 25\) and \(13x - 19\) are alternate exterior and interior angles (equal).

$$\begin{align*} 9x+25&=13x-19\\ 25 + 19&=13x-9x\\ 44&=4x\\ x&=11 \end{align*}$$

Now, if \(13x-19\) and \(17y + 5\) are vertical angles (equal). Substitute \(x = 11\) into \(13x-19\), we have \(13\times11-19=124\). Then \(17y+5=124\)

$$\begin{align*} 17y&=124 - 5\\ 17y&=119\\ y&=7 \end{align*}$$

But if we assume the options are given. Let's check \(x = 11\)
For \(x = 11\), \(9x+25=9\times11 + 25=99+25 = 124\), \(13x-19=13\times11-19=143-19 = 124\)
If \(17y+5=124\), then \(y = 7\) (but there is no such option). Wait, maybe there is a mis - interpretation of angles.
If \(9x + 25\) and \(17y + 5\) are corresponding angles (equal) and \(13x-19\) and \(9x + 25\) are vertical angles (equal)
\(9x+25=13x-19\), \(x = 11\)
\(9x+25=9\times11+25=124\), then \(17y + 5=124\), \(y = 7\) (not in options).
Wait, if we assume \(9x+25\) and \(13x - 19\) are same - side interior angles (\(9x+25+13x-19=180\), \(22x+6 = 180\), \(x=\frac{174}{22}\approx7.91\) wrong)
Another approach: check the options
Option 1: \(x = 11\), \(y = 3\)
\(9x+25=9\times11+25=124\), \(13x-19=13\times11-19 = 124\), \(17y+5=17\times3+5=56\) (wrong)
Option 2: \(x = 12\), \(y = 9\)
\(9x+25=9\times12+25=108 + 25=133\), \(13x-19=13\times12-19=156-19 = 137\) (wrong)
Option 3: \(x = 5\), \(y = 14\)
\(9x+25=9\times5+25=45 + 25=70\), \(13x-19=13\times5-19=65-19 = 46\) (wrong)

Wait, there is a mistake in the problem's options or in the angle - relationship assumption. Let's re - assume the angle relationship:
If \(9x + 25\) and \(17y + 5\) are alternate exterior and interior angles (equal) and \(13x-19\) and \(9x + 25\) are vertical angles (equal)
\(9x+25=13x-19\), \(x = 11\)
If \(9x+25=17y + 5\) (corresponding angles)
Substitute \(x = 11\)
\(9\times11+25=17y+5\)
\(99 + 25-5=17y\)
\(119=17y\)
\(y = 7\) (not in options)

If we assume \(13x-19\) and \(17y + 5\) are supplementary (same - side interior angles) and \(9x + 25=13x-19\) (vertical angles)
\(x = 11\)
\(13x-19=124\), then \(17y+5=180 - 124=56\), \(y=\frac{56 - 5}{17}=\frac{51}{17}=3\)

Answer:

\(x = 11,y = 3\)