Write an equation where the expression for the exterior angle at $R$, $(5x+5)^\circ$, is equal to the sum of the expressions for $\angle P$ and $\angle Q$, $(3x+8)^\circ$ and $(x+27)^\circ$.

After you have the equation, first combine like terms on the right side, then move all the $x$ terms to one side and the constants to the other side to solve for $x$.

In the first line, enter y1 = 5x + 5 to represent the exterior angle. In the second line, enter y2 = 3x + 8 + x + 27 (or simplify it to y2 = 4x + 35) to represent the sum of the two remote interior angles.

Look at the graph of $y_1$ and $y_2$ and tap on their point of intersection. The x-coordinate of this point is the value of $x$ that makes the exterior angle equal to the sum of the two interior angles.

You can also test each option by creating a table: add a table for one of the expressions, type the answer choices (18, 24, 30, 36) into the $x$-column, and compare the corresponding $y$-values for $5x+5$ and $3x+8+x+27$. The correct choice is the one where the two values match exactly.

In a triangle, an exterior angle is equal to the sum of the two interior angles that are not adjacent to it (the two "remote" interior angles).

Here, the exterior angle at $R$ is $(5x+5)^\circ$, and the two remote interior angles are $\angle P = (3x+8)^\circ$ and $\angle Q = (x+27)^\circ$.

So we know:

Start from the relationship:

Combine like terms on the right side:

• Combine the $x$ terms: $3x + x = 4x$
• Combine the constants: $8 + 27 = 35$

So the equation becomes:

Solve the equation:

1. Subtract $4x$ from both sides:

2. Subtract $5$ from both sides:

So the value of $x$ is $30$, which corresponds to choice C) 30.