Question

1. given that m || n, solve for x and y. given that m || n, solve for x and y. 24. given that m || n, solve for x and y. 25. given that m || n, solve for x and y.

Explanation:

1. For problem 24:

• Since \(m\parallel n\), we can use the property of corresponding - angles or alternate - interior angles. Here, \((4x - 10)^{\circ}\) and \((8x-2)^{\circ}\) are corresponding angles (or we can use the property of linear - pair and parallel - lines relationships). If they are corresponding angles, they are equal.
• Step 1: Set up the equation for \(x\)
• \(4x-10 = 8x - 2\).
• Subtract \(4x\) from both sides: \(-10=8x - 2-4x\), which simplifies to \(-10 = 4x-2\).
• Add 2 to both sides: \(-10 + 2=4x\), so \(-8 = 4x\).
• Divide both sides by 4: \(x=-2\).
• Step 2: Set up the equation for \(y\)
• We know that \((2y)^{\circ}\) and \((4x - 10)^{\circ}\) are supplementary (a linear - pair). First, substitute \(x = - 2\) into \(4x-10\): \(4\times(-2)-10=-8 - 10=-18\). Then, since \(2y+(4x - 10)=180\) (linear - pair), substituting \(x=-2\) gives \(2y-18 = 180\).
• Add 18 to both sides: \(2y=180 + 18=198\).
• Divide both sides by 2: \(y = 99\).

1. For problem 25:

• Since \(m\parallel n\), \((8x + 1)^{\circ}\) and \((9x-16)^{\circ}\) are corresponding angles (or we can use other angle - relationships based on parallel lines).
• Step 1: Set up the equation for \(x\)
• \(8x + 1=9x-16\).
• Subtract \(8x\) from both sides: \(1=9x-16-8x\), so \(x=17\).
• Step 2: Set up the equation for \(y\)
• We know that \((4y + 3)^{\circ}\) and \((8x + 1)^{\circ}\) are either corresponding, alternate - interior, or related by some other parallel - line angle property. Let's assume they are corresponding angles. First, substitute \(x = 17\) into \(8x + 1\): \(8\times17+1=136 + 1=137\).
• Then, since \(4y+3=8x + 1\), substituting \(x = 17\) gives \(4y+3=137\).
• Subtract 3 from both sides: \(4y=137 - 3=134\).
• Divide both sides by 4: \(y=\frac{134}{4}=33.5\).

Answer:

• Problem 24: \(x=-2,y = 99\)
• Problem 25: \(x = 17,y=33.5\)