Question

1. which two theorems would justify that ( mangle4 = mangle6 ), given that ( mangle5 = mangle6 ) in the diagram below?

right angles theorem, exterior angles theorem
vertical angles theorem, alternate interior angles theorem
corresponding angles theorem, angle addition theorem
vertical angles theorem, consecutive interior angles theorem

Explanation:

• Vertical Angles Theorem: Vertical angles are equal. So, \(m\angle4 = m\angle5\) because \(\angle4\) and \(\angle5\) are vertical angles.
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, alternate - interior angles are equal. Here, if we assume the horizontal lines are parallel (implied by the angle - relationship context), and given \(m\angle5=m\angle6\) (from the problem statement), we use the fact that \(m\angle4 = m\angle5\) (vertical angles) and then substitute \(m\angle5\) with \(m\angle6\) (since \(m\angle5 = m\angle6\)) to get \(m\angle4=m\angle6\).

Answer:

vertical angles theorem, alternate interior angles theorem