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$In /_\PQR, bar(PR) is extended through point R to point S,m/_RPQ=(3x+15)^(@), m/_PQR=(2x-9)^(@), and m/_QRS=(7x-12)^(@). Find m/_RPQ. Answer:$

In $\triangle \mathrm{PQR}, \overline{P R}$ is extended through point $\mathrm{R}$ to point $\mathrm{S}, \mathrm{m} \angle R P Q=(3 x+15)^{\circ}$ , $\mathrm{m} \angle P Q R=(2 x-9)^{\circ}$ , and $\mathrm{m} \angle Q R S=(7 x-12)^{\circ}$ . Find $\mathrm{m} \angle R P Q$ . $\newline$ Answer:

Full solution

Q. In

\triangle \mathrm{PQR}, \overline{P R}

is extended through point

\mathrm{R}

to point

\mathrm{S}, \mathrm{m} \angle R P Q=(3 x+15)^{\circ}

\mathrm{m} \angle P Q R=(2 x-9)^{\circ}

, and

\mathrm{m} \angle Q R S=(7 x-12)^{\circ}

. Find

\mathrm{m} \angle R P Q

\newline

Answer:

Write Equation: To solve for the measure of angle $RPQ$ , we need to use the fact that the sum of the measures of the angles in a straight line is $180$ degrees. This means that the sum of the measures of angles $PQR$ and $QRS$ must equal $180$ degrees.
Combine Like Terms: First, let's write the equation that represents the sum of the measures of angles PQR and QRS: $\newline$ $m\angle PQR + m\angle QRS = 180^\circ$ $\newline$ Substitute the given expressions for $m\angle PQR$ and $m\angle QRS$ : $\newline$ $(2x - 9)^\circ + (7x - 12)^\circ = 180^\circ$
Isolate Terms with x: Combine like terms to solve for x: $\newline$ $2x - 9 + 7x - 12 = 180$ $\newline$ $9x - 21 = 180$
Solve for x: Add $21$ to both sides of the equation to isolate the terms with $x$ : $\newline$ $9x - 21 + 21 = 180 + 21$ $\newline$ $9x = 201$
Substitute $x$ into Expression: Divide both sides by $9$ to solve for $x$ : $\frac{9x}{9} = \frac{201}{9}$ $x = 22.33$ (repeating)
Recalculate Measure of Angle: Now that we have the value of $x$ , we can find the measure of angle RPQ by substituting $x$ back into the expression for $m\angle RPQ$ : $m\angle RPQ = (3x + 15)^\circ$ $m\angle RPQ = (3(22.33) + 15)^\circ$ $m\angle RPQ = (66.99 + 15)^\circ$ $m\angle RPQ = 81.99^\circ$ (repeating)
Identify Mistake: However, we have made a mistake. The measure of an angle should be a whole number, and we cannot have a repeating decimal for the measure of an angle. We need to recheck our calculations.