Question

given:
\\( \overline { uh } \perp \overline { uz } \\).
\\( \overline { ub } \perp \overline { ug } \\).
\\( m \angle hub = 20 ^ { \circ } \\).
enter the measures and reasons to justify each
needed to calculate \\( m \angle z u g \\).
\\( m \angle z u g = \\)
diagram notes:

Explanation:

Step1: Use the property of perpendicular lines

Since \( \overline{UH}\perp\overline{UZ}\), \(m\angle HUZ = 90^{\circ}\). By the angle - addition postulate, \(m\angle HUB+m\angle BUZ=m\angle HUZ\). Given \(m\angle HUB = 20^{\circ}\), then \(m\angle BUZ=m\angle HUZ - m\angle HUB\). Substituting the values, \(m\angle BUZ=90^{\circ}- 20^{\circ}=70^{\circ}\).

Step2: Use the property of perpendicular lines again

Since \( \overline{UB}\perp\overline{UG}\), \(m\angle BUG = 90^{\circ}\). By the angle - addition postulate, \(m\angle BUZ+m\angle ZUG=m\angle BUG\). We know \(m\angle BUG = 90^{\circ}\) and \(m\angle BUZ = 70^{\circ}\). Then \(m\angle ZUG=m\angle BUG - m\angle BUZ\).

Step3: Calculate \(m\angle ZUG\)

Substitute \(m\angle BUG = 90^{\circ}\) and \(m\angle BUZ = 70^{\circ}\) into the formula \(m\angle ZUG=m\angle BUG - m\angle BUZ\). So \(m\angle ZUG=90^{\circ}-70^{\circ}=20^{\circ}\).

Answer:

\(20\)