Question

in the figure below, ( mangle1=(x - 23)^circ ) and ( mangle2 = 6x^circ ). find the angle measures.
( mangle1=square^circ )
( mangle2=square^circ )

Explanation:

Step1: Use the property of adjacent angles forming a linear pair

Since \(\angle1\) and \(\angle2\) form a linear pair, \(m\angle1 + m\angle2=180^{\circ}\).
Substitute \(m\angle1=(x - 23)^{\circ}\) and \(m\angle2 = 6x^{\circ}\) into the equation: \((x-23)+6x=180\).

Step2: Solve the equation for \(x\)

Combine like - terms: \(x+6x-23 = 180\), which simplifies to \(7x-23=180\).
Add 23 to both sides: \(7x=180 + 23\), so \(7x=203\).
Divide both sides by 7: \(x=\frac{203}{7}=29\).

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

Substitute \(x = 29\) into \(m\angle1=(x - 23)^{\circ}\): \(m\angle1=(29-23)^{\circ}=6^{\circ}\).
Substitute \(x = 29\) into \(m\angle2 = 6x^{\circ}\): \(m\angle2=6\times29^{\circ}=174^{\circ}\).

Answer:

\(m\angle1 = 6^{\circ}\), \(m\angle2=174^{\circ}\)