Question

in the figure below, ( mangle1=(x - 24)^{circ} ) and ( mangle2 = 5x^{circ} ). find the angle measures.
( mangle1=square^{circ} )
( mangle2=square^{circ} )

Explanation:

Step1: Use the property of adjacent angles

Since \(\angle1\) and \(\angle2\) are adjacent angles forming a straight - line, \(m\angle1 + m\angle2=180^{\circ}\). Given \(m\angle1=(x - 24)^{\circ}\) and \(m\angle2 = 5x^{\circ}\), we have the equation \((x - 24)+5x=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(x+5x-24 = 180\), which simplifies to \(6x-24 = 180\). Add 24 to both sides: \(6x=180 + 24\), so \(6x=204\). Divide both sides by 6: \(x=\frac{204}{6}=34\).

Step3: Find \(m\angle1\) and \(m\angle2\)

Substitute \(x = 34\) into the expressions for the angles.
For \(m\angle1\): \(m\angle1=(x - 24)^{\circ}=(34 - 24)^{\circ}=10^{\circ}\).
For \(m\angle2\): \(m\angle2=5x^{\circ}=5\times34^{\circ}=170^{\circ}\).

Answer:

\(m\angle1 = 10^{\circ}\), \(m\angle2=170^{\circ}\)