Question

find the values of the variables and the measures of the angles. 29. 30.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. For $\triangle PQR$, we have $(2x - 9)+(2x + 4)+x=180$.

Step2: Combine like - terms

Combining the $x$ terms and the constant terms, we get $(2x+2x + x)+(-9 + 4)=180$, which simplifies to $5x-5 = 180$.

Step3: Solve for $x$

Add 5 to both sides of the equation: $5x-5 + 5=180 + 5$, so $5x=185$. Then divide both sides by 5: $x=\frac{185}{5}=37$.

Step4: Find the angle measures

For $\angle P=(2x - 9)^{\circ}$, substitute $x = 37$: $\angle P=(2\times37-9)^{\circ}=(74 - 9)^{\circ}=65^{\circ}$.
For $\angle Q=(2x + 4)^{\circ}$, substitute $x = 37$: $\angle Q=(2\times37 + 4)^{\circ}=(74+4)^{\circ}=78^{\circ}$.
For $\angle R=x^{\circ}$, $\angle R = 37^{\circ}$.

Answer:

$x = 37$, $\angle P=65^{\circ}$, $\angle Q=78^{\circ}$, $\angle R = 37^{\circ}$