Question

complete the proof that ∠s = ∠q.

(there is a diagram with triangles and segments marked with congruency ticks, and a table with statements and reasons:)

statement | reason
--- | ---

1. ( overline{tu} cong overline{vw} ) | given
2. ( overline{rv} cong overline{ru} ) | given
3. ( overline{st} cong overline{qw} ) | given
4. ( overline{rs} cong overline{qr} ) | given
5. ( qu = qr + ru ) | blank
6. ( sv = rs + rv ) | additive property of length
7. ( qu = rs + rv ) | substitution
8. ( qu = sv ) | transitive property of equality
9. ( uw = vw + uv ) | additive property of length
10. ( tv = tu + uv ) | additive property of length
11. ( uw = tu + uv ) | substitution
12. ( tv = uw ) | blank
13. ( \triangle quw cong \triangle svt ) | sss
14. ( angle s = angle q ) | cpctc

Explanation:

Step1: Identify the property for step 5

The statement \( QU = QR + RU \) is formed by adding the lengths of \( QR \) and \( RU \) to get \( QU \). This is the Additive Property of Length, which states that if a segment is composed of two smaller segments, the length of the whole segment is the sum of the lengths of the smaller ones.

Step2: Confirm the reason

Since \( QU \) is made up of \( QR \) and \( RU \) (as seen in the diagram, \( R \) is a point on \( QU \) such that \( QR + RU = QU \)), the reason for \( QU = QR + RU \) is the Additive Property of Length.

Answer:

Additive Property of Length