Question

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Explanation:

Step1: Analyze ∠WVX and ∠TVU

We know \( \overrightarrow{SV} \) bisects \( \angle TVX \), and \( \angle W \cong \angle U \). Also, from the diagram (and the fact that vertical angles or the structure here), \( \angle WVX \) and \( \angle TVU \) are vertical angles? Wait, no, actually, since \( \overrightarrow{SV} \) bisects \( \angle TVX \), and looking at the angles formed, \( \angle WVX \) and \( \angle TVU \) are vertical angles? Wait, no, more accurately, they are vertical angles (opposite angles formed by intersecting lines) or actually, since \( \angle W \cong \angle U \), and the lines form these angles, but actually, the reason for \( \angle WVX \cong \angle TVU \) is that they are vertical angles, so Vertical Angles Theorem (Vertical angles are congruent). Wait, but also, from the given that \( \angle W \cong \angle U \), but no, step 3: \( \angle WVX \cong \angle TVU \). Wait, actually, \( \angle WVX \) and \( \angle TVU \) are vertical angles (formed by the intersection of \( WU \) and \( X T \) at \( V \)), so by Vertical Angles Congruence Theorem, they are congruent. Alternatively, since \( \overrightarrow{SV} \) bisects \( \angle TVX \), but step 3: let's check. Wait, the reason for \( \angle WVX \cong \angle TVU \) is that they are vertical angles, so Vertical Angles are congruent. So the reason is "Vertical Angles Congruence Theorem" (or "Vertical angles are congruent").

Step2: Confirm the Reason

So for statement 3, \( \angle WVX \cong \angle TVU \), the reason is that vertical angles are congruent (Vertical Angles Congruence Theorem).

For step 7: \( m\angle SVW = m\angle SVT + m\angle TVU \). From step 3, \( \angle WVX \cong \angle TVU \), so \( m\angle WVX = m\angle TVU \) (Definition of Congruent Angles). Then from step 5: \( m\angle SVW = m\angle SVX + m\angle WVX \), and step 4: \( \angle SVX \cong \angle SVT \) (so \( m\angle SVX = m\angle SVT \)). So substituting \( m\angle WVX \) with \( m\angle TVU \) and \( m\angle SVX \) with \( m\angle SVT \) in step 5, we get \( m\angle SVW = m\angle SVT + m\angle TVU \). So the reason for step 7 is "Substitution Property" (since we substitute \( m\angle WVX \) with \( m\angle TVU \) and \( m\angle SVX \) with \( m\angle SVT \) from steps 3 and 4).

Wait, the original problem has step 3 and step 7 to fill. Let's focus on step 3 first. The question is to complete the proof, so for step 3: \( \angle WVX \cong \angle TVU \), the reason is "Vertical Angles are congruent" (Vertical Angles Congruence Theorem).

For step 7: \( m\angle SVW = m\angle SVT + m\angle TVU \), the reason is "Substitution Property" (because from step 3, \( \angle WVX \cong \angle TVU \) implies \( m\angle WVX = m\angle TVU \), and from step 4, \( \angle SVX \cong \angle SVT \) implies \( m\angle SVX = m\angle SVT \), then substitute into step 5: \( m\angle SVW = m\angle SVX + m\angle WVX = m\angle SVT + m\angle TVU \)).

But the user's problem: let's check the table. The user has step 3 and step 7 with empty reasons. Let's assume the first empty is step 3. So step 3: \( \angle WVX \cong \angle TVU \), reason: Vertical Angles are congruent (Vertical Angles Congruence Theorem).

Answer:

For step 3, the reason is "Vertical Angles are congruent" (or "Vertical Angles Congruence Theorem"). For step 7, the reason is "Substitution Property" (using \( \angle WVX \cong \angle TVU \) and \( \angle SVX \cong \angle SVT \) to substitute into the angle sum). But if we focus on step 3 (the first empty), the answer is that \( \angle WVX \cong \angle TVU \) because vertical angles are congruent, so the reason is "Vertical Angles Congruence Theorem" (or "Vertical angles are congruent").