Question

which of the following does not represent possible measures for < pqr, < rqs, and < pqs?

m∠pqr = x + 5, m∠rqs = 2x + 5, m∠pqs = x²

11°, 25° and 36°
10°, 15°, and 25°
1°, 3° and 4°
all of these are correct.

Explanation:

Step1: Use angle - addition postulate

Since $\angle PQS=\angle PQR+\angle RQS$, we have $x^{2}=(x + 5)+(2x + 5)$.

Step2: Simplify the equation

$x^{2}=x + 5+2x + 5$, which simplifies to $x^{2}-3x - 10 = 0$.

Step3: Factor the quadratic equation

$(x - 5)(x+2)=0$.

Step4: Solve for x

Setting each factor equal to zero gives $x - 5=0$ or $x + 2=0$. So $x = 5$ or $x=-2$. Since angle measures are non - negative, we consider $x = 5$.
When $x = 5$, $m\angle PQR=x + 5=10^{\circ}$, $m\angle RQS=2x + 5=15^{\circ}$, and $m\angle PQS=x^{2}=25^{\circ}$.

Answer:

The measures $10^{\circ},15^{\circ}$, and $25^{\circ}$ are possible.
For $m\angle PQR = 11^{\circ}$, then $x+5 = 11$, so $x = 6$. And $m\angle RQS=2x + 5=2\times6 + 5=17^{\circ}$, $m\angle PQS=x^{2}=36^{\circ}$, but $11 + 17
eq36$.
For $m\angle PQR = 1^{\circ}$, then $x + 5=1$, so $x=-4$. And $m\angle RQS=2x + 5=2\times(-4)+5=-3^{\circ}$ (not possible as angle measure is non - negative).
The answer is $11^{\circ},25^{\circ}$ and $36^{\circ}$ and $1^{\circ},3^{\circ}$ and $4^{\circ}$.