Question

1. given that ( mangle lkn = 145^{circ} ), find ( mangle lkm ) and ( mangle mkn ).

Explanation:

Step1: Set up the equation

Since \(m\angle LKN=(2x + 10)^{\circ}+(4x-3)^{\circ}\) and \(m\angle LKN = 145^{\circ}\), we have \((2x + 10)+(4x-3)=145\).

Step2: Simplify the left - hand side

Combine like terms: \(2x+4x+10 - 3=145\), which simplifies to \(6x + 7=145\).

Step3: Solve for \(x\)

Subtract \(7\) from both sides: \(6x=145 - 7=138\). Then divide both sides by \(6\): \(x=\frac{138}{6}=23\).

Step4: Find \(m\angle LKM\)

Substitute \(x = 23\) into \(2x + 10\). So \(m\angle LKM=2\times23+10=46 + 10=56^{\circ}\).

Step5: Find \(m\angle MKN\)

Substitute \(x = 23\) into \(4x-3\). So \(m\angle MKN=4\times23-3=92-3=89^{\circ}\).

Answer:

\(m\angle LKM = 56^{\circ}\), \(m\angle MKN=89^{\circ}\)