18) Notice that angles (11y-32)° and (6x+7)° are opposite with respect to the vertex, then they are congruent.
[tex](11y-32)=(6x+7)[/tex]
On the other hand, consider any of the angles that are formed by the same vertex as (11y-32)° and (6x+7)°. Any of those is congruent with the angle (3x-16)° (One by the corresponding relation and the other by being alternate exterior angles).
Therefore:
[tex](3x-16)+(6x+7)=180[/tex]
With those two equations, we can solve for x and y as follows:
[tex]\begin{cases}(3x-16)+(6x+7)=180 \\ (11y-32)-(6x+7)=0\end{cases}[/tex][tex]\begin{gathered} \Rightarrow9x-9=180 \\ \Rightarrow x=21 \end{gathered}[/tex]
Therefore, x=21°
As for y:
[tex]\begin{gathered} \Rightarrow(11y-32)-(6(21)+7)=0 \\ \Rightarrow11y-165=0 \\ \Rightarrow y=15 \end{gathered}[/tex]
Then, y=15°
19) Notice that angles (8x-14)° and (5y+16)° are supplementary, then:
[tex](8x-14)+(5y+16)=180[/tex]
And angles (8x-14)° and (5x+34)° are alternate exterior angles, thus they are congruent:
[tex]\begin{gathered} (8x-14)=(5x+34) \\ \Rightarrow(8x-14)-(5x+34)=0 \end{gathered}[/tex]
The system of equations is:
[tex]\begin{gathered} \begin{cases}(8x-14)+(5y+16)=180 \\ (8x-14)-(5x+34)=0\end{cases} \\ \end{gathered}[/tex]
[tex]\begin{gathered} \Rightarrow(8x-14)-(5x+34)=0 \\ \Rightarrow3x-48=0 \\ \Rightarrow x=16 \end{gathered}[/tex]
x=16°
[tex]\begin{gathered} \Rightarrow(8(16)-14)+(5y+16)=180 \\ \Rightarrow5y+130=180 \\ \Rightarrow y=10 \end{gathered}[/tex]
y=10°
20) Notice that angles (5y-23)° and (3x)° are generated by the same transversal line and that they are corresponding angles. Therefore,
[tex]\begin{gathered} (5y-23)=3x \\ \Rightarrow(5y-23)-3x=0 \end{gathered}[/tex]
Notice that angles (2x+13)° and (3x+47)° are supplementary, as the figure below shows:
The angles in the same color are congruent.
Then, we get the following system of equations:
[tex]\begin{cases}(5y-23)-3x=0 \\ (2x+13)+(3x+47)=180\end{cases}[/tex]
Solving for x and y:
[tex]\begin{gathered} (2x+13)+(3x+47)=180 \\ \Rightarrow5x+60=180 \\ \Rightarrow x=24 \end{gathered}[/tex]
The answer is x=24°
[tex]\begin{gathered} (5y-23)-3x=0 \\ \Rightarrow(5y-23)-3(24)=0 \\ \Rightarrow5y-95=0 \\ \Rightarrow y=19 \end{gathered}[/tex]
The answer is y=19°
