Question

find ( mangle k ) in parallelogram ( hijk ).
( mangle k=square^{circ} )

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, consecutive angles are supplementary. So, \((t + 52)+(3t-92)=180\).

Step2: Solve the equation for \(t\)

Combine like terms: \(t+3t+52 - 92=180\), \(4t-40 = 180\). Add \(40\) to both sides: \(4t=180 + 40=220\). Divide by \(4\): \(t=\frac{220}{4}=55\).

Step3: Find \(m\angle H\)

Substitute \(t = 55\) into \(t + 52\), \(m\angle H=55+52 = 107^{\circ}\).

Step4: Use the property of parallelogram (opposite angles are equal)

In parallelogram \(HIJK\), \(\angle K\) and \(\angle J\) are adjacent to \(\angle H\). Also, \(\angle K\) and \(\angle H\) are consecutive angles. Since consecutive angles in a parallelogram are supplementary, or we can use the fact that \(\angle K\) and \(\angle I\) are adjacent (\(\angle H\) and \(\angle I\) are also consecutive). Another way: Since \(\angle H\) and \(\angle K\) are consecutive, \(m\angle K=180-(t + 52)\). After \(t = 55\), \(m\angle K=180-(55 + 52)=73^{\circ}\).

Answer:

\(73\)