Question

use the diagram to find the measures of all the angles, given that ( mangle1 = 76^{circ} ).( mangle2=square^{circ} )( mangle3=square^{circ} )( mangle4=square^{circ} )

Explanation:

Step1: Use the linear - pair relationship

Angles \( \angle1\) and \( \angle2\) form a linear pair. The sum of angles in a linear pair is \(180^{\circ}\).
So, \(m\angle1 + m\angle2=180^{\circ}\).
Given \(m\angle1 = 76^{\circ}\), then \(m\angle2=180^{\circ}-m\angle1\).
Substitute \(m\angle1 = 76^{\circ}\) into the formula: \(m\angle2 = 180^{\circ}-76^{\circ}=104^{\circ}\).

Step2: Use the vertical - angle relationship

Vertical angles are equal. \( \angle1\) and \( \angle3\) are vertical angles.
So, \(m\angle3=m\angle1\).
Since \(m\angle1 = 76^{\circ}\), then \(m\angle3 = 76^{\circ}\).

Step3: Use the vertical - angle relationship

\( \angle2\) and \( \angle4\) are vertical angles.
So, \(m\angle4=m\angle2\).
Since \(m\angle2 = 104^{\circ}\), then \(m\angle4 = 104^{\circ}\).

Answer:

\(m\angle2 = 104^{\circ}\), \(m\angle3 = 76^{\circ}\), \(m\angle4 = 104^{\circ}\)