Question

in δnop, \\(\overline{np}\\) is extended through point p to point q, \\(\text{m}\angle nop = (3x + 19)\degree\\), \\(\text{m}\angle pno = (3x + 8)\degree\\), and \\(\text{m}\angle opq = (8x + 15)\degree\\). find \\(\text{m}\angle pno\\).

answer attempt 1 out of 2

\\(\text{m}\angle pno = \square\degree\\)

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Explanation:

Step1: Use Exterior Angle Theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures 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: $(8x + 15)=(3x + 19)+(3x + 8)$

Step2: Solve for x

Simplify the right - hand side of the equation:
$(3x + 19)+(3x + 8)=3x+3x + 19 + 8=6x+27$

So our equation becomes $8x + 15=6x+27$

Subtract $6x$ from both sides: $8x-6x + 15=6x-6x+27$
$2x+15 = 27$

Subtract 15 from both sides: $2x+15 - 15=27 - 15$
$2x=12$

Divide both sides by 2: $x = 6$

Step3: 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:

$26$