If 𝘮∠𝘕𝘖𝘗=𝟤𝘹+𝟤, 𝘮∠𝘗𝘖𝘙=𝟥𝘹−𝟧,m normal angle upper P upper O upper R times equals 3 times sans-serif-italic x minus 5 comma and 𝘮∠𝘕𝘖𝘘=𝟣𝟣𝟦, what is the value of x?

Given the angles \( m\angle NOP = 2x + 2 \), \( m\angle POR = 3x - 5 \), and \( m\angle NOQ = 144^\circ \), we are asked to find the value of \( x \).

Assuming that points \( N \), \( O \), \( P \), \( Q \), and \( R \) are on a straight line or form a linear pair (since \( m\angle NOQ = 144^\circ \)), we can deduce that the angles \( m\angle NOP \) and \( m\angle POR \) must sum up to \( m\angle NOQ \). Thus, we have the equation:

\[
m\angle NOP + m\angle POR = m\angle NOQ
\]

Substituting the given expressions for the angles:

\[
(2x + 2) + (3x - 5) = 144
\]

Now, let's simplify and solve for \( x \):

\[
2x + 3x + 2 - 5 = 144
\]

\[
5x - 3 = 144
\]

Add 3 to both sides:

\[
5x = 147
\]

Finally, divide by 5:

\[
x = \frac{147}{5} = 29.4
\]

So, the value of \( x \) is **29.4**.

Would you like a detailed explanation or have any questions?

Here are 5 related questions:
1. What is the measure of \( m\angle NOP \) when \( x = 29.4 \)?
2. How would you check if the angles sum correctly when \( x = 29.4 \)?
3. Can the value of \( x \) be negative in this scenario?
4. What assumptions are made about the relationship between points \( N \), \( O \), \( P \), \( Q \), and \( R \)?
5. How would the problem change if \( m\angle NOQ \) was different?

**Tip:** Always verify your final answer by substituting it back into the original equations to ensure all conditions are met.

Explore how to solve angles in a straight line problem involving geometric concepts and linear pairs. Given specific angle measurements and relationships, find the value of x using algebraic equations. Suitable for Grades 9-12 students.

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