Given the problem:

• $m∠NOP = 2x + 2$
• $m∠POR = 3x - 5$
• $m∠NOQ = 114^\circ$

It appears that angles $∠NOP$ and $∠POR$ are part of the same angle $∠NOQ$, which has a measure of $114^\circ$. This implies that:

$m∠NOP + m∠POR = m∠NOQ$

We can now set up the equation:

$(2x + 2) + (3x - 5) = 114$

Let's simplify and solve for $x$:

$2x + 3x + 2 - 5 = 114$ $5x - 3 = 114$ $5x = 117$ $x = \frac{117}{5} = 23.4$

Now that we have $x = 23.4$, we can substitute it back into the expressions for $m∠NOP$ and $m∠POR$ to find the measures of these angles:

1. $m∠NOP = 2x + 2 = 2(23.4) + 2 = 46.8 + 2 = 48.8^\circ$
2. $m∠POR = 3x - 5 = 3(23.4) - 5 = 70.2 - 5 = 65.2^\circ$

Thus:

• $m∠NOP = 48.8^\circ$
• $m∠POR = 65.2^\circ$
• $m∠NOQ = 114^\circ$ (which checks out since $48.8 + 65.2 = 114$).

Would you like more details or have any further questions?

Here are 5 related questions:

1. What are the types of angles formed by the sum of $m∠NOP$ and $m∠POR$?
2. How can we verify if these angles form a linear pair?
3. Can we apply the exterior angle theorem here?
4. What would happen if $m∠NOQ$ was a right angle?
5. How would the solution change if $m∠POR$ was doubled?

Tip: Always check if the sum of angles equals a given total when angles are adjacent.