Question

find the values of the variables and the measures of the angles. 29. 30. 31

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. For triangle PQR, we have the equation $(2x - 9)+(2x + 4)+x=180$.

Step2: Combine like - terms

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

Step3: Isolate the variable 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 measures of the angles

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

Answer:

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