Question

1. if (mangle pqr = 141^{circ}), find each measure. (x=)(mangle pqs=)(mangle sqr=)

Explanation:

Step1: Set up equation based on angle - sum

Since \(m\angle PQR=m\angle PQS + m\angle SQR\), we have \((13x + 4)+(10x-1)=141\).

Step2: Combine like - terms

Combine the \(x\) terms and the constant terms on the left - hand side: \(13x+10x + 4-1=141\), which simplifies to \(23x+3 = 141\).

Step3: Isolate the variable term

Subtract 3 from both sides of the equation: \(23x=141 - 3\), so \(23x=138\).

Step4: Solve for \(x\)

Divide both sides by 23: \(x=\frac{138}{23}=6\).

Step5: Find \(m\angle PQS\)

Substitute \(x = 6\) into the expression for \(m\angle PQS\): \(m\angle PQS=13x + 4=13\times6+4=78 + 4=82^{\circ}\).

Step6: Find \(m\angle SQR\)

Substitute \(x = 6\) into the expression for \(m\angle SQR\): \(m\angle SQR=10x-1=10\times6-1=59^{\circ}\).

Answer:

\(x = 6\)
\(m\angle PQS=82^{\circ}\)
\(m\angle SQR=59^{\circ}\)