Question

select the correct answer. given ( angle woa cong angle aoz ) prove ( angle boy cong angle xob ) since ( wy ), ( xz ), and ( overline{ab} ) intersect at ( o ), ( angle woa cong angle boy ) and ( angle aoz cong angle xob ) by the vertical angles theorem. it is given that ( angle woa cong angle aoz ). by the transitive property, ( angle boy cong angle xob ). what is the conclusion reached by the proof? a. ( angle aoz cong angle xob ) b. ( angle boy cong angle xob ) c. ( angle woa cong angle aoz ) d. ( angle woa cong angle boy )

Explanation:

Step1: Analyze Given Information

We know \(\angle WOA \cong \angle BOY\) (given), \(\angle WOA \cong \angle AOZ\) (by some property), and \(\angle BOY \cong \angle XOB\) (by vertical angles theorem). We need to find the conclusion from the proof.

Step2: Use Transitive Property of Congruence

The transitive property of congruence states that if \(a \cong b\) and \(b \cong c\), then \(a \cong c\). Here, \(\angle WOA \cong \angle BOY\) and \(\angle BOY \cong \angle XOB\), so by transitive property, \(\angle WOA \cong \angle XOB\)? Wait, no, wait. Wait, also \(\angle WOA \cong \angle AOZ\) and \(\angle BOY \cong \angle XOB\), and \(\angle WOA \cong \angle BOY\). Wait, let's re - examine the options.

Wait, the given is \(\angle WOA \cong \angle AOZ\) (let's call this statement 1), \(\angle WOA \cong \angle BOY\) (given), and \(\angle BOY \cong \angle XOB\) (vertical angles). We need to find the conclusion. Let's check the options:

Option A: \(\angle AOZ \cong \angle XOB\). Since \(\angle WOA \cong \angle AOZ\) (statement 1) and \(\angle WOA \cong \angle BOY\) (given) and \(\angle BOY \cong \angle XOB\) (vertical angles), by transitive property, \(\angle AOZ \cong \angle XOB\) (because \(\angle AOZ \cong \angle WOA\) and \(\angle WOA \cong \angle BOY\) and \(\angle BOY \cong \angle XOB\), so \(\angle AOZ \cong \angle XOB\)).

Option B: \(\angle BOY \cong \angle XOB\) is given (vertical angles), not a conclusion from the proof steps combining all.

Option C: \(\angle WOA \cong \angle AOZ\) is given (by some property), not a conclusion from the whole proof.

Option D: \(\angle WOA \cong \angle BOY\) is given, not a conclusion from the proof steps combining.

So the correct conclusion is \(\angle AOZ \cong \angle XOB\) which is option A.

Answer:

A. \(\angle AOZ \cong \angle XOB\)