Question

question
in $\triangle nop$, $\overline{np}$ is extended through point $p$ to point $q$, $m\angle pno = (3x + 12)\degree$, $m\angle nop = (3x + 15)\degree$, and $m\angle opq = (8x + 17)\degree$. what is the value of $x$?

Explanation:

Step1: Apply the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
In \(\triangle NOP\), \(\angle OPQ\) is an exterior angle. So, \(m\angle PNO+m\angle NOP = m\angle OPQ\).
Substitute the given angle expressions: \((3x + 12)+(3x + 15)=8x + 17\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(3x+3x + 12 + 15=8x + 17\).
\(6x+27 = 8x + 17\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(6x - 6x+27=8x - 6x + 17\).
\(27 = 2x+17\).
Subtract \(17\) from both sides: \(27 - 17=2x+17 - 17\).
\(10 = 2x\).
Divide both sides by \(2\): \(x=\frac{10}{2}\).

Answer:

\(x = 5\)