Question

1. given that m || n, solve for x and y. 25. given that m || n, solve for x and y.

Explanation:

Step1: Identify angle - relationships

Since \(m\parallel n\), corresponding - angles are equal. Let's assume that the pairs of angles given are corresponding or alternate - interior/exterior angles. For problem 24, if we assume that \((3x + 17)^{\circ}\) and \((5x-19)^{\circ}\) are corresponding or alternate - interior/exterior angles, then they are equal. So we set up the equation \(3x + 17=5x-19\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(17 = 5x-3x-19\), which simplifies to \(17 = 2x-19\). Then add 19 to both sides: \(17 + 19=2x\), so \(36 = 2x\). Divide both sides by 2, we get \(x = 18\).
Let's assume that \((3y + 4)^{\circ}\) and \(90^{\circ}\) are corresponding or alternate - interior/exterior angles (assuming a right - angle situation). Then \(3y+4 = 90\). Subtract 4 from both sides: \(3y=90 - 4=86\), and \(y=\frac{86}{3}\).

For problem 25, assume that \((9x-14)^{\circ}\) and \((6x + 4)^{\circ}\) are corresponding or alternate - interior/exterior angles. Set up the equation \(9x-14=6x + 4\).

Step3: Solve the equation for \(x\) in problem 25

Subtract \(6x\) from both sides: \(9x-6x-14=4\), which simplifies to \(3x-14 = 4\). Add 14 to both sides: \(3x=4 + 14=18\). Divide both sides by 3, we get \(x = 6\).
If we assume that \(y\) and \((9x-14)\) are supplementary (a common angle - relationship for parallel lines), and since \(x = 6\), then \(9x-14=9\times6-14=54 - 14 = 40\). If \(y\) and \((9x - 14)\) are supplementary, then \(y=180-(9x - 14)\). Substituting \(x = 6\), we get \(y = 140\).

Answer:

Problem 24: \(x = 18\), \(y=\frac{86}{3}\)
Problem 25: \(x = 6\), \(y = 140\)