Question

complete the proof that $\triangle tuy \cong \triangle wxv$.
image of triangles tuy and wxv with markings

	statement	reason
1	$\overline{tu} \cong \overline{wx}$	given
2	$\overline{vw} \cong \overline{ty}$	given
3	$\overline{uv} \cong \overline{xy}$	given
4	$uy = xy + ux$	additive property of length
5	$vx = uv + ux$	additive property of length
6	$uy = uv + ux$	dropdown for reason
7	$vx = uy$	dropdown with options: reflexive property of congruence, reflexive property of equality, substitution, transitive property of congruence, transitive property of equality
8	$\triangle tuy \cong \triangle wxv$	dropdown for reason

submit

Explanation:

Step1: Analyze Step 6's Reason

We know from Step 3 that \( \overline{UV} \cong \overline{XY} \), so \( UV = XY \) (since congruent segments have equal length). In Step 4, \( UY = XY + UX \), and we substitute \( XY \) with \( UV \) (from Step 3's congruence leading to equality) to get \( UY = UV + UX \). This is the Substitution property, as we substitute \( XY \) with its equal \( UV \).

Step2: Analyze Step 7's Reason

From Step 5, \( VX = UV + UX \), and from Step 6, \( UY = UV + UX \). So we can say \( VX = UY \) because if two quantities are both equal to the same quantity (\( UV + UX \)), they are equal to each other. This is the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \); here \( a = VX \), \( b = UV + UX \), \( c = UY \)).

Step3: Analyze Step 8's Reason

We have \( \overline{TU} \cong \overline{WX} \) (Step 1), \( \overline{TY} \cong \overline{VW} \) (Step 2), and now \( \overline{UY} \cong \overline{VX} \) (since \( VX = UY \) implies congruence of segments). So by the SSS (Side - Side - Side) Congruence Criterion, \( \triangle TUY \cong \triangle WXV \). But for the dropdown in Step 6, we saw that Step 6 uses Substitution (substituting \( XY \) with \( UV \) in the length equation), Step 7 uses Transitive Property of Equality, and for the proof of triangle congruence, after establishing all three sides congruent, SSS is used. But focusing on the dropdown for Step 6:
Looking at Step 6: We have \( UY = XY + UX \) (Step 4) and \( UV = XY \) (from Step 3, \( \overline{UV} \cong \overline{XY} \) implies \( UV = XY \)). So we substitute \( XY \) with \( UV \) in Step 4's equation to get \( UY = UV + UX \). So the reason for Step 6 is Substitution. For Step 7, since \( VX = UV + UX \) (Step 5) and \( UY = UV + UX \) (Step 6), by Transitive Property of Equality, \( VX = UY \). For Step 8, since \( TU \cong WX \), \( TY \cong VW \), and \( UY \cong VX \), by SSS, \( \triangle TUY \cong \triangle WXV \). But the question is about completing the proof, and for the dropdowns:

For Step 6: The reason is Substitution (we substitute \( XY \) with \( UV \) because \( UV = XY \) from Step 3).

For Step 7: The reason is Transitive Property of Equality (as both \( VX \) and \( UY \) equal \( UV + UX \)).

For Step 8: The reason is SSS (Side - Side - Side) Congruence Criterion, but since the options given for Step 7's dropdown (and maybe Step 6) - let's re - check the dropdown for Step 6:

Wait, the user's question is to complete the proof. Let's re - examine the table:

Step 6: Statement is \( UY = UV + UX \), Reason is to be chosen. We know from Step 3 that \( UV = XY \) (since \( \overline{UV} \cong \overline{XY} \)). Step 4 is \( UY = XY + UX \). So we substitute \( XY \) with \( UV \) in Step 4 to get Step 6. So the reason for Step 6 is Substitution.

Step 7: Statement is \( VX = UY \), Reason: From Step 5, \( VX = UV + UX \); from Step 6, \( UY = UV + UX \). So by Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \); here \( a = VX \), \( b = UV + UX \), \( c = UY \)), so the reason is Transitive Property of Equality.

Step 8: Statement is \( \triangle TUY \cong \triangle WXV \), Reason: SSS (since \( TU \cong WX \), \( TY \cong VW \), \( UY \cong VX \)). But the dropdowns for Step 6: the options are Reflexive Property of Congruence, Reflexive Property of Equality, Substitution, Transitive Property of Congruence, Transitive Property of Equality.

So for Step 6: Substitution (because we substitute \( XY \) with \( UV \) in the length equation).

For Step 7: Transitive Property of Equality (because \( VX \) and \( UY \) are both equal to \( UV + UX \)).

For Step 8: SSS (but since it's not in the dropdown for Step 6, focusing on the dropdowns given:

Let's answer the parts:

For Step 6 Reason: Substitution

For Step 7 Reason: Transitive Property of Equality

For Step 8 Reason: SSS (Side - Side - Side) Congruence Criterion

But the question is to complete the proof. Let's assume the user is asking about the dropdown for Step 6 first.

Answer:

For Step 6: Substitution

For Step 7: Transitive Property of Equality

For Step 8: SSS (Side - Side - Side) Congruence Criterion

(If we focus on the dropdown for Step 6, the answer is Substitution; for Step 7, Transitive Property of Equality; for the triangle congruence, SSS)