Question

complete the proof that ∠s ≅ ∠q.
(image of a geometric figure with triangles and segments marked with congruency ticks, and a two - column proof table with statements and reasons:

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. $qu = qr + ru$ -
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. $tv = uw$ -
13. $\triangle quw \cong \triangle svt$ - sss
14. $\angle s \cong \angle q$ - cpctc)

Explanation:

Step1: Analyze Statement 5

Statement 5 is \( QU = QR + RU \). This is using the Additive Property of Length, which states that if a segment is composed of two smaller segments, its length is the sum of the lengths of the smaller ones. Here, \( QU \) is composed of \( QR \) and \( RU \), so the reason is the Additive Property of Length.

Step2: Analyze Statement 12

Statement 12 is \( TV = UW \). From statements 11 (\( UW = TU + UV \)) and 10 (\( TV = TU + UV \)), by the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \)), since \( UW = TU + UV \) and \( TV = TU + UV \), we can conclude \( TV = UW \). So the reason is the Transitive Property of Equality.

Answer:

• For Statement 5: Additive Property of Length
• For Statement 12: Transitive Property of Equality