Question

prove that $\triangle tuv \cong \triangle twv$.
image of quadrilateral tuvw with diagonal tv, markings indicating tu ≅ tw, uv ≅ wv, and tv is common side
statement reason
1 $tv \cong tv$ pick reason
2 $tu \cong tw$ given
3 $uv \cong wv$ pick reason
4 $\triangle tuv \cong \triangle twv$ pick congruence criterion

Explanation:

Step1: Identify Reason for \( TV \cong TV \)

The segment \( TV \) is congruent to itself, which is the Reflexive Property of Congruence. So the reason for \( TV \cong TV \) is "Reflexive Property of Congruence".

Step2: Identify Reason for \( UV \cong WV \)

From the diagram, the marks on \( UV \) and \( WV \) indicate they are given as congruent (or maybe from the diagram's markings, but typically in such problems, if marked with same tick marks, it's "Given" or "Markings on the diagram indicate congruence", but since it's a pick reason, and the diagram has equal marks on \( UV \) and \( WV \), we can say "Given (from diagram markings)" or more precisely, if it's marked, the reason is that the segments with the same number of tick marks are congruent, so "Marked segments are congruent" or "Given (from the diagram's congruence markings)". But more formally, if it's part of the given (from the diagram's tick marks), the reason is that those segments are marked as congruent, so we can say "Given (diagram shows \( UV \cong WV \))" or "Congruent segments (marked with same tick marks)".

Step3: Identify Congruence Criterion for \( \triangle TUV \cong \triangle TWV \)

We have \( TU \cong TW \) (given), \( UV \cong WV \) (given or from markings), and \( TV \cong TV \) (reflexive). So by the Side - Side - Side (SSS) Congruence Criterion, the triangles are congruent.

Answer:

1. For \( TV \cong TV \): Reflexive Property of Congruence
2. For \( UV \cong WV \): Marked segments are congruent (or Given from diagram markings)
3. For \( \triangle TUV \cong \triangle TWV \): SSS (Side - Side - Side) Congruence Criterion