Question

△uvw and △stw are equilateral. complete the proof that △swv ≅ △tuw.
statement
1 △uvw is equilateral
2 △stw is equilateral
3 (overline{uw} cong overline{vw})
4 (overline{sw} cong overline{tw})
5 (angle swv cong angle twu)
6 △swv ≅ △tuw
reason
1
2
3
4
5
6

Explanation:

Step1: State given premise

Given (Problem statement)

Step2: State given premise

Given (Problem statement)

Step3: Use equilateral triangle property

All sides of equilateral $\triangle UVW$ are congruent, so $\overline{UW} \cong \overline{VW}$.

Step4: Use equilateral triangle property

All sides of equilateral $\triangle STW$ are congruent, so $\overline{SW} \cong \overline{TW}$.

Step5: Use vertical angles theorem

Vertical angles formed by intersecting lines are congruent, so $\angle SWV \cong \angle TWU$.

Step6: Apply SAS congruence rule

We have $\overline{SW} \cong \overline{TW}$, $\angle SWV \cong \angle TWU$, $\overline{VW} \cong \overline{UW}$, so by SAS (Side-Angle-Side) congruence postulate, $\triangle SVW \cong \triangle TUW$.

Answer:

Statement	Reason
2. $\triangle STW$ is equilateral	Given
3. $\overline{UW} \cong \overline{VW}$	All sides of equilateral $\triangle UVW$ are congruent
4. $\overline{SW} \cong \overline{TW}$	All sides of equilateral $\triangle STW$ are congruent
5. $\angle SWV \cong \angle TWU$	Vertical Angles Theorem
6. $\triangle SVW \cong \triangle TUW$	SAS Congruence Postulate