Question

1. $mangle fkl=x + 156$, $mangle jkf=x + 33$, and $mangle jkl = 167^{circ}$. find $mangle fkl$.
2. $mangle pqr=55x + 2$, $mangle pqf=23x + 1$, and $mangle fqr = 65^{circ}$. find $mangle pqf$.

Explanation:

Step1: Set up the equation

Since \(m\angle PQR=m\angle PQF + m\angle FQR\), we have the equation \(55x + 2=(23x + 1)+65\).

Step2: Simplify the right - hand side

\(55x + 2=23x+66\).

Step3: Move the \(x\) terms to one side

Subtract \(23x\) from both sides: \(55x-23x + 2=23x-23x + 66\), which gives \(32x+2 = 66\).

Step4: Move the constant to the other side

Subtract 2 from both sides: \(32x+2 - 2=66 - 2\), so \(32x=64\).

Step5: Solve for \(x\)

Divide both sides by 32: \(x=\frac{64}{32}=2\).

Step6: Find \(m\angle PQF\)

Substitute \(x = 2\) into the expression for \(m\angle PQF\): \(m\angle PQF=23x + 1=23\times2+1=46 + 1=47^{\circ}\).

Answer:

\(m\angle PQF = 47^{\circ}\)