Question

△stw and △uvw are equilateral. complete the proof that \\(\overline{tu} \cong \overline{sv}\\).

	statement	reason
2	△uvw is equilateral
3	\\(\overline{sw} \cong \overline{tw}\\)
4	\\(\overline{uw} \cong \overline{vw}\\)
5	\\(\angle swv \cong \angle twu\\)
6	△svw ≅ △tuw
7	\\(\overline{tu} \cong \overline{sv}\\)

Explanation:

Step1: Reason for Statement 1

Given (as per problem statement, $\triangle STW$ is equilateral)

Step2: Reason for Statement 2

Given (as per problem statement, $\triangle UVW$ is equilateral)

Step3: Reason for Statement 3

All sides of an equilateral triangle are congruent (in $\triangle STW$, $SW$ and $TW$ are sides, so $SW \cong TW$)

Step4: Reason for Statement 4

All sides of an equilateral triangle are congruent (in $\triangle UVW$, $UW$ and $VW$ are sides, so $UW \cong VW$)

Step5: Reason for Statement 5

$\angle SWV$ and $\angle TWU$ are both $60^\circ + \angle UW S$ (or vertical angles with added $60^\circ$ from equilateral triangles, so they are congruent by angle addition or since each equilateral triangle has $60^\circ$ angles and the included angle is same)

Step6: Reason for Statement 6

SAS Congruence ( $SW \cong TW$, $UW \cong VW$, $\angle SWV \cong \angle TWU$, so $\triangle SVW \cong \triangle TUW$ by SAS)

Step7: Reason for Statement 7

Corresponding parts of congruent triangles are congruent (CPCTC, since $\triangle SVW \cong \triangle TUW$, $TU$ and $SV$ are corresponding parts, so $TU \cong SV$)

Answer:

1. Given
2. Given
3. All sides of an equilateral triangle are congruent
4. All sides of an equilateral triangle are congruent
5. $\angle SWV$ and $\angle TWU$ are each $60^\circ + \angle UWS$ (or vertical angles with equal added angles, so congruent)
6. SAS Congruence ( $SW \cong TW$, $UW \cong VW$, $\angle SWV \cong \angle TWU$ )
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)