Question

complete the proof that $\triangle uvw \cong \triangle xst$.
(image of two triangles with markings and a table with statements and reasons:

1. $\overline{tu} \cong \overline{wx}$ (given)
2. $\angle uvw \cong \angle tsx$ (given)
3. $\angle vuw \cong \angle sxt$ (given)
4. $tx = wx + tw$ (additive property of length)
5. $uw = tu + tw$ ( blank for reason )
6. $tx = tu + tw$ ( blank for reason )
7. $uw = tx$ (transitive property of equality)
8. $\triangle uvw \cong \triangle xst$ ( blank for reason ))

Explanation:

Step1: Analyze Statement 5

To find the reason for \( UW = TU + TW \), we observe the segment addition. From the diagram, \( UW \) is composed of \( TU \) and \( TW \), so this is by the Additive Property of Length (Segment Addition Postulate).

Step2: Analyze Statement 6

For \( TX = TU + TW \), we use the Substitution Property. Since \( \overline{TU} \cong \overline{WX} \) (so \( TU = WX \)) and from Statement 4 \( TX = WX + TW \), we substitute \( WX \) with \( TU \), so the reason is Substitution Property of Equality.

Step3: Analyze Statement 8

To prove \( \triangle UVW \cong \triangle XST \), we have two angles and the included side. We know \( \angle UVW \cong \angle TSX \), \( \angle VUW \cong \angle SXT \), and \( UW = TX \) (included side). So by the Angle - Side - Angle (ASA) Congruence Postulate, \( \triangle UVW \cong \triangle XST \).

Filling in the Blanks:

• Statement 5 Reason: Additive Property of Length (Segment Addition Postulate)
• Statement 6 Reason: Substitution Property of Equality
• Statement 8 Reason: ASA (Angle - Side - Angle) Congruence Postulate

Answer:

• Statement 5 Reason: Additive Property of Length
• Statement 6 Reason: Substitution Property of Equality
• Statement 8 Reason: ASA (Angle - Side - Angle) Congruence Postulate