Question

complete the proof that \\(\overline{rw} \cong \overline{tu}\\).

(image of two triangles with marked angles and segments, and a table with statements and reasons:

1. \\(\angle rws \cong \angle utv\\) (given)
2. \\(\angle tuv \cong \angle srw\\) (given)
3. \\(\overline{vw} \cong \overline{st}\\) (given)
4. \\(sw = vw + sv\\) (reason to be filled)
5. \\(tv = st + sv\\) (additive property of length)
6. \\(sw = st + sv\\) (substitution)
7. \\(tv = sw\\) (reason to be filled)
8. \\(\triangle rsw \cong \triangle uvt\\) (reason to be filled)
9. \\(\overline{rw} \cong \overline{tu}\\) (cpctc))

Explanation:

Step1: Analyze Step 4

The Additive Property of Length states that if a point lies on a segment, the length of the segment is the sum of the lengths of its parts. Here, \( V \) is on \( SW \), so \( SW = VW + SV \) by the Additive Property of Length.

Step2: Analyze Step 7

From Step 5 (\( TV = ST + SV \)) and Step 6 (\( SW = ST + SV \)), we can use the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \)) to conclude \( TV = SW \).

Step3: Analyze Step 8

We have \( \angle RWS \cong \angle UTV \) (Step 1), \( \angle TUV \cong \angle SRW \) (Step 2), and \( TV = SW \) (Step 7). By the Angle - Angle - Side (AAS) Congruence Criterion, \( \triangle RSW \cong \triangle UVT \).

Answer:

• Step 4 Reason: Additive Property of Length
• Step 7 Reason: Transitive Property of Equality
• Step 8 Reason: AAS (Angle - Angle - Side) Congruence Criterion