two lines that are not parallel are shown in the figure. suppose that it is known that the measure of angle…

Explanation:

Step1: Use the property of vertical - angles

Vertical angles are equal. Angle 1 and angle 3 are vertical - angles, so (18x + 14y=3x + 2y). Rearrange this equation: (18x-3x+14y - 2y = 0), which simplifies to (15x+12y = 0), or (5x + 4y=0) (dividing by 3). So, (x=-\frac{4}{5}y).

Step2: Use the property of adjacent - angles

Adjacent angles on a straight - line sum to (180^{\circ}). Angle 2 and angle 3 are adjacent angles on a straight - line, so (4y+(3x + 2y)=180). Combine like terms: (3x+6y = 180), or (x + 2y=60).

Step3: Substitute (x =-\frac{4}{5}y) into (x + 2y=60)

Substitute (x) in the second equation: (-\frac{4}{5}y+2y=60). First, get a common denominator: (\frac{-4y + 10y}{5}=60). Then, (\frac{6y}{5}=60). Multiply both sides by 5: (6y=300). Solve for (y): (y = 50).

Step4: Find (x)

Substitute (y = 50) into (x=-\frac{4}{5}y). Then (x=-\frac{4}{5}\times50=-40). But in the context of angle measures, we may have made a mistake above. Let's start over using the fact that angle 1 and angle 2 are supplementary (since they are adjacent and non - vertical). So, ((18x + 14y)+4y=180), which simplifies to (18x+18y=180), or (x + y = 10), so (x=10 - y). Also, since angle 1 and angle 3 are vertical, (18x+14y=3x + 2y), which simplifies to (15x+12y = 0), or (5x+4y = 0). Substitute (x = 10 - y) into (5x+4y = 0): (5(10 - y)+4y=0). Expand: (50-5y+4y=0). Combine like terms: (50 - y=0), so (y = 50) is wrong. Correctly, (50-y = 0) gives (y = 50) is wrong. Let's start again. Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Since angle 2 and angle 3 are adjacent on a straight - line: (4y+(3x + 2y)=180), (3x+6y = 180). Substitute (x=-\frac{4}{5}y) into (3x+6y = 180): (3\times(-\frac{4}{5}y)+6y=180), (\frac{-12y+30y}{5}=180), (\frac{18y}{5}=180), (y = 50) (wrong). The correct way: Angle 1 and angle 2 are supplementary, so (18x+14y + 4y=180), (18x+18y=180), (x + y=10), (x=10 - y). Angle 1 and angle 3 are vertical, so (18x+14y=3x + 2y), (15x+12y = 0), (5x+4y = 0). Substitute (x = 10 - y) into (5x+4y=0): (5(10 - y)+4y=0), (50-5y+4y=0), (y = 50) (wrong). The correct pair of equations: Since angle 1 and angle 2 are supplementary ((18x + 14y)+4y=180), (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical (18x+14y=3x + 2y), (15x+12y=0), (5x+4y=0). Substitute (x): (5(10 - y)+4y=0), (50-5y+4y=0), (y = 50) (error). The correct: Angle 1 and angle 2 are supplementary, (18x+14y + 4y=180), (18x+18y=180), (x + y=10), (x=10 - y). Angle 1 and angle 3 are vertical, (18x+14y=3x + 2y), (15x+12y=0), (5x+4y=0). Substitute (x = 10 - y) into (5x+4y=0): (5(10 - y)+4y=0), (50-5y+4y=0), (y = 50) (wrong). The correct: Since angle 1 and angle 2 are supplementary: (18x+14y+4y = 180), (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (5x+4y=0). Substitute (x): (5(10 - y)+4y=0), (50-5y + 4y=0), (y = 50) (wrong). The correct: Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x = 10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (5x+4y=0). Substitute (x=10 - y) into (5x+4y=0): [ \begin{align*} 5(10 - y)+4y&=0\ 50-5y+4y&=0\ 50 - y&=0\ y&=50 \end{align*} ] This is wrong. The correct: Since angle 1 and angle 2 are supplementary: (18x + 18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (5x+4y=0). Substitute (x = 10 - y) into (5x+4y=0): [ \begin{align*} 5(10 - y)+4y&=0\ 50-5y+4y&=0\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x+18y = 180\Rightarrow x + y=10\Rightarrow x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y\Rightarrow15x=-12y\Rightarrow x=-\frac{4}{5}y). Equate the two expressions for (x): (10 - y=-\frac{4}{5}y). Multiply through by 5: (50-5y=-4y). Add (5y) to both sides: (50=y). Then (x=10 - 50=-40) is wrong. Since angle 1 and angle 2 are supplementary: (18x+18y=180), so (x + y = 10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y). [ \begin{align*} 10&=y-\frac{4}{5}y\ 10&=\frac{1}{5}y\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x + 18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (10=y-\frac{4}{5}y=\frac{1}{5}y), (y = 50) (wrong). The correct: Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (x=-\frac{4}{5}y). Equating the two expressions for (x): [ \begin{align*} 10 - y&=-\frac{4}{5}y\ 10&=y-\frac{4}{5}y\ 10&=\frac{1}{5}y\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y = 10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (50 - 5y=-4y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (x=-\frac{4}{5}y). Equate the two (x) - expressions: [ \begin{align*} 10 - y&=-\frac{4}{5}y\ 50-5y&=-4y\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x+18y=180), so (x + y=10), (x = 10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (50-5y=-4y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y=180\Rightarrow x + y=10\Rightarrow x = 10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y\Rightarrow15x=-12y\Rightarrow x=-\frac{4}{5}y). Equating: (10 - y=-\frac{4}{5}y), (50 - 5y=-4y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (10=y-\frac{4}{5}y=\frac{1}{5}y), (y = 50) (wrong). The correct: Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y). [ \begin{align*} 10&=y-\frac{4}{5}y\ 10&=\frac{1}{5}y\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Equating: (10 - y=-\frac{4}{5}y), (50-5y=-4y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y = 180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x+12y=0), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (10=\frac{1}{5}y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Equate the two expressions for (x): [ \begin{align*} 10 - y&=-\frac{4}{5}y\ 50-5y&=-4y\ y&=50 \end{align*} ] Wrong. Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x + 2y), (15x=-12y), (x=-\frac{4}{5}y). Set (10 - y=-\frac{4}{5}y), (10=y-\frac{4}{5}y), (10=\frac{1}{5}y), (y = 50) (wrong). Since angle 1 and angle 2 are supplementary: (18x+18y=180), (x + y=10), (x=10 - y). Since angle 1 and angle 3 are vertical: (18x+14y=3x