Question

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question
in δnop, \\(\overline{np}\\) is extended through point p to point q, \\(m\angle nop = (3x + 19)\degree\\), \\(m\angle pno = (3x + 8)\degree\\), and \\(m\angle opq = (8x + 15)\degree\\). find \\(m\angle pno\\).
answer attempt 1 out of 2
\\(m\angle pno = \square\degree\\) submit answer

Explanation:

Step1: Recall 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 $m\angle OPQ=m\angle NOP + m\angle PNO$.

Substitute the given angle measures into the formula:
$$8x + 15=(3x + 19)+(3x + 8)$$

Step2: Simplify the right - hand side of the equation

First, combine like terms on the right - hand side:
$$(3x+3x)+(19 + 8)=6x+27$$
So our equation becomes:
$$8x + 15=6x+27$$

Step3: Solve for $x$

Subtract $6x$ from both sides of the equation:
$$8x-6x + 15=6x-6x+27$$
$$2x+15 = 27$$
Then subtract 15 from both sides:
$$2x+15 - 15=27 - 15$$
$$2x=12$$
Divide both sides by 2:
$$x=\frac{12}{2}=6$$

Step4: Find $m\angle PNO$

We know that $m\angle PNO=(3x + 8)^{\circ}$. Substitute $x = 6$ into the expression:
$$m\angle PNO=3\times6+8=18 + 8=26$$

Answer:

$26$