Question

question
in $\triangle pqr$, $\overline{pr}$ is extended through point $r$ to point $s$, $m\angle pqr = (3x + 13)\degree$, $m\angle qrs = (7x + 7)\degree$, and $m\angle rpq = (x + 12)\degree$. find $m\angle rpq$.
answer attempt 1 out of 2
$m\angle rpq = \square$
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Explanation:

Step1: Recall the 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 $\triangle PQR$, $\angle QRS$ is an exterior angle, so $m\angle QRS=m\angle PQR + m\angle RPQ$.

Step2: Substitute the given angle measures

We know that $m\angle QRS=(7x + 7)^{\circ}$, $m\angle PQR=(3x + 13)^{\circ}$, and $m\angle RPQ=(x + 12)^{\circ}$. Substituting these into the equation from the Exterior Angle Theorem, we get:
$$7x+7=(3x + 13)+(x + 12)$$

Step3: Simplify the right - hand side of the equation

Simplify the right - hand side: $(3x + 13)+(x + 12)=3x+x+13 + 12=4x+25$. So our equation becomes:
$$7x+7 = 4x+25$$

Step4: Solve for $x$

Subtract $4x$ from both sides: $7x-4x+7=4x - 4x+25$, which simplifies to $3x+7 = 25$. Then subtract 7 from both sides: $3x+7 - 7=25 - 7$, so $3x=18$. Divide both sides by 3: $x=\frac{18}{3}=6$.

Step5: Find $m\angle RPQ$

We know that $m\angle RPQ=(x + 12)^{\circ}$. Substitute $x = 6$ into this expression: $m\angle RPQ=(6 + 12)^{\circ}=18^{\circ}$.

Answer:

$18$