Question

y (7x + 11)° (16x - 38)°

Explanation:

1. First, assume the two - angle expressions represent vertical angles (since no other information about the relationship between the angles is given, and vertical angles are equal when two lines intersect).

• Step 1: Set up the equation
• Vertical angles are equal. So, we set \(7x + 11=16x - 38\).
• Step 2: Solve the equation for \(x\)
• Subtract \(7x\) from both sides of the equation: \(7x+11 - 7x=16x - 38-7x\).
• This simplifies to \(11 = 9x-38\).
• Then add 38 to both sides: \(11 + 38=9x-38 + 38\).
• We get \(49 = 9x\).
• Divide both sides by 9: \(x=\frac{49}{9}\).
• Step 3: Find the value of \(y\)
• Since \(y\) and \((7x + 11)\) are adjacent angles and form a straight - line (linear pair), \(y+(7x + 11)=180\).
• First, find the value of \(7x + 11\) when \(x = \frac{49}{9}\).
• \(7x+11=7\times\frac{49}{9}+11=\frac{343}{9}+\frac{99}{9}=\frac{343 + 99}{9}=\frac{442}{9}\).
• Then \(y=180-\frac{442}{9}=\frac{1620}{9}-\frac{442}{9}=\frac{1620 - 442}{9}=\frac{1178}{9}\approx130.89\).

1. If we assume the two non - \(y\) angles are supplementary (form a straight - line):

• Step 1: Set up the equation
• \((7x + 11)+(16x - 38)=180\).
• Step 2: Combine like terms
• \(7x+16x+11 - 38=180\).
• \(23x-27 = 180\).
• Step 3: Solve for \(x\)
• Add 27 to both sides: \(23x-27 + 27=180 + 27\).
• \(23x=207\).
• Divide both sides by 23: \(x = 9\).
• Step 4: Find the value of \(y\)
• If \(x = 9\), then one of the non - \(y\) angles, say \(7x + 11=7\times9+11=63 + 11=74\).
• Since \(y\) and this angle are vertical angles, \(y = 74\).

Since the most common relationship assumed for two non - \(y\) angles formed by two intersecting lines is that they are supplementary, we take the second approach.

Step1: Set up angle - sum equation

\((7x + 11)+(16x - 38)=180\)

Step2: Combine like terms

\(23x-27 = 180\)

Step3: Isolate \(x\) term

\(23x=180 + 27=207\)

Step4: Solve for \(x\)

\(x=\frac{207}{23}=9\)

Step5: Find \(y\) value

Since \(y\) and \(7x + 11\) are vertical angles, when \(x = 9\), \(7x+11=7\times9+11 = 74\), so \(y = 74\)

Answer:

\(x = 9,y = 74\)