Question

compare the proof with the diagram.

(diagram: a geometric figure with points t, u, s, v, w, q, r. some sides have congruency marks.)

statement	reason
2. \\( \overline{st} \cong \overline{qw} \\)	given
3. \\( \overline{vw} \cong \overline{tu} \\)	given
4. \\( \overline{rv} \cong \overline{ru} \\)	given
5. \\( qu = qr + ru \\)	additive property of length
6. \\( sv = rs + rv \\)	additive property of length
7. \\( qu = rs + rv \\)	substitution
8. \\( qu = sv \\)	transitive property of equality
9. \\( uw = vw + uv \\)
10. \\( tv = tu + uv \\)	additive property of length
11. \\( uw = tu + uv \\)
12. \\( tv = uw \\)	transitive property of equality
13. \\( \triangle stv \cong \triangle qwu \\)

Explanation:

Step1: Identify the Property

The statement in step 9 is \( UW = VW + UV \). We need to find the reason, which should be the Additive Property of Length, as it's the same as step 5, 6, and 10 where the additive property is used to express a segment as the sum of two smaller segments.

Step2: Confirm the Property

The Additive Property of Length states that if a point lies on a segment, the length of the whole segment is the sum of the lengths of the two smaller segments. Here, \( V \) is a point on \( UW \) (implied by the diagram and the segment addition), so \( UW = VW + UV \) follows the Additive Property of Length.

For step 11, since \( VW = TU \) (given in step 3), we substitute \( VW \) with \( TU \) in the equation from step 9 (\( UW = VW + UV \)) to get \( UW = TU + UV \), so the reason is Substitution.

For step 13, we have three sides of \( \triangle STV \) and \( \triangle QWU \) equal: \( ST = QW \) (step 2), \( SV = QU \) (step 8), \( TV = UW \) (step 12). So by the Side - Side - Side (SSS) Congruence Postulate, \( \triangle STV\cong\triangle QWU \).

Step 9 Reason:

Step1: Recall Additive Property

The Additive Property of Length says that for a segment with a point in between, the total length is the sum of the parts. Here, \( UW \) is split by \( V \) into \( VW \) and \( UV \), so \( UW = VW + UV \) uses the Additive Property of Length.
<The reason is Additive Property of Length>

Step 11 Reason:

Step1: Identify Substitution

We know \( VW = TU \) (step 3). From step 9, \( UW = VW + UV \). Substituting \( VW \) with \( TU \) gives \( UW = TU + UV \), so the reason is Substitution.
<The reason is Substitution>

Step 13 Reason:

Step1: Check SSS Congruence

We have \( ST = QW \) (step 2), \( SV = QU \) (step 8), \( TV = UW \) (step 12). By the SSS (Side - Side - Side) Congruence Postulate, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. So \( \triangle STV\cong\triangle QWU \) by SSS.
<The reason is SSS Congruence Postulate>

Answer:

s:

• Step 9 Reason: Additive Property of Length
• Step 11 Reason: Substitution
• Step 13 Reason: SSS Congruence Postulate