d) Recall that by the exterior angle theorem we know that:
[tex]a+b=d\text{.}[/tex]Therefore,
[tex]5x+10+6x+2=12x\text{.}[/tex]Adding like terms we get:
[tex]11x+12=12x\text{.}[/tex]Subtracting 11x from both sides of the equation we get:
[tex]\begin{gathered} 11x+12-11x=12x-11x, \\ 12=12x-11x, \\ x=12. \end{gathered}[/tex]Therefore, we get that:
[tex]d=12x=12(12)=144.[/tex]Recall that c and d are a linear pair, meaning:
[tex]\measuredangle c+\measuredangle d=180^{\circ}.[/tex]Substituting d= 180 degrees and solving for angle c we get:
[tex]\measuredangle c=180^{\circ}-144^{\circ}=36^{\circ}.[/tex]Answer part d):
[tex]\begin{gathered} x=12, \\ \measuredangle c=36^{\circ}. \end{gathered}[/tex]c) To solve this question we will use the fact that the interior angles of a triangle add up to 180 degrees, and that h and i are a linear pair.
Since the interior angles of a triangle add up to 180 degrees, then:
[tex]\measuredangle k+\measuredangle j+\measuredangle h=180^{\circ}.[/tex]Solving the above equation for angle h we get:
[tex]\measuredangle h=180^{\circ}-\measuredangle j-\measuredangle k.[/tex]Substituting ∡j=42°, ∡k=50° we get:
[tex]\measuredangle h=88^{\circ}.[/tex]Now, since angles h and i are a linear pair, then:
[tex]\measuredangle h+\measuredangle i=180^{\circ}.[/tex]Solving for angle i we get:
[tex]\measuredangle i=180^{\circ}-\measuredangle h=180^{\circ}-88^{\circ}=92^{\circ}.^{}[/tex]Finally, we know that:
[tex]\measuredangle i+\measuredangle n+\measuredangle m=180^{\circ}.[/tex]Substituting the measures of angles i, and n, and solving for m we get:
[tex]\measuredangle m=180^{\circ}-92^{\circ}-33^{\circ}=55^{\circ}.^{}[/tex]Answer part c):
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