Question

1. in triangle pqr, m∠p=(10x - 4)°, m∠q=(4x - 22)°, m∠r=(x + 11)°. x = _, m∠p = _, m∠q = _, m∠r = _

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, $(10x - 4)+(4x - 22)+(x + 11)=180$.

Step2: Combine like - terms

$(10x+4x + x)+(-4-22 + 11)=180$, which simplifies to $15x-15 = 180$.

Step3: Add 15 to both sides

$15x-15 + 15=180 + 15$, resulting in $15x=195$.

Step4: Divide both sides by 15

$\frac{15x}{15}=\frac{195}{15}$, so $x = 13$.

Step5: Find the measure of each angle

For $\angle R$: Substitute $x = 13$ into $x + 11$, we get $13+11=24$.
For $\angle Q$: Substitute $x = 13$ into $4x - 22$, we get $4\times13-22=52 - 22 = 30$.
For $\angle P$: Substitute $x = 13$ into $10x - 4$, we get $10\times13-4=130 - 4 = 126$.

Answer:

$x = 13$
$m\angle P=126^{\circ}$
$m\angle Q = 30^{\circ}$
$m\angle R=24^{\circ}$