[tex]ANSWER:
\begin{gathered} \small \begin{array}{l} 1.\: \bold{Given:}\: \textsf{Triangles }DEF\textsf{ and }GHI\textsf{ such that} \\ \qquad\qquad\:\triangle DEF \cong \triangle GHI \\ \\ \quad\bold{Prove:}\: \triangle GHI \cong \triangle DEF \\ \\ \quad\bold{Proof:} \end{array} \end{gathered}1.Given:Triangles DEF and GHI such that△DEF≅△GHIProve:△GHI≅△DEFProof:
\begin{gathered} \scriptsize \begin{array}{| l | l |} \hline \quad\quad\quad \text{STATEMENTS} & \quad\quad\quad\!\text{REASONS} \\ \hline \triangle DEF \cong \triangle GHI & \quad\quad\quad\quad\!\! \textsf{Given} \\ \hline \scriptsize {\!\!\begin{array}{l} \overline{DE}\cong \overline{GH};\: \overline{EF}\cong \overline{HI};\: \overline{DF}\cong \overline{GI} \\ \angle D \cong \angle G;\: \angle E \cong \angle H;\: \angle F \cong \angle I\end{array}} \negthickspace & \negthickspace {\!\!\begin{array}{c}1.\textsf{ Corresponding Parts of} \\ \:\:\:\textsf{Congruent Triangles are} \\ \:\:\: \textsf{Congruent (CPCTC)} \end{array}} \\ \hline \scriptsize{\!\!\begin{array}{l} \overline{GH}\cong \overline{DE};\: \overline{HI}\cong \overline{EF};\: \overline{GI}\cong \overline{DF} \\ \angle G \cong \angle D;\: \angle H \cong \angle E;\: \angle I \cong \angle F \!\! \end{array}} & {\!\!\begin{array}{c} 2.\textsf{ Symmetric Property of} \\ \:\:\:\textsf{Congruence}\end{array}\!\!} \\ \hline \triangle GHI \cong \triangle DEF & {\begin{array}{c} \textsf{Definition of Congruent}\\ \textsf{Triangles}\end{array}} \\ \hline \end{array} \\ \: \end{gathered}STATEMENTS△DEF≅△GHIDE≅GH;EF≅HI;DF≅GI∠D≅∠G;∠E≅∠H;∠F≅∠IGH≅DE;HI≅EF;GI≅DF∠G≅∠D;∠H≅∠E;∠I≅∠F△GHI≅△DEFREASONSGiven1. Corresponding Parts ofCongruent Triangles areCongruent (CPCTC)2. Symmetric Property ofCongruenceDefinition of CongruentTriangles
\begin{gathered} \small \begin{array}{l} 2.\: \bold{Given:}\: \overline{AB}\cong \overline{CD},\: \overline{AB} \perp \overline{BD},\: \overline{CD} \perp \overline{BD}, \\ \qquad\qquad\: E\textsf{ is the midpoint of }\overline{BD} \\ \\ \quad\bold{Prove:}\: \triangle ABE \cong \triangle CDE \\ \\ \quad \bold{Proof:} \end{array} \end{gathered}2.Given:AB≅CD,AB⊥BD,CD⊥BD,E is the midpoint of BDProve:△ABE≅△CDEProof:
\begin{gathered} \scriptsize \begin{array}{| l | l |} \hline \quad \text{STATEMENTS} & \qquad\qquad \text{REASONS} \\ \hline {\begin{array}{l} \overline{AB} \cong \overline{CD} \\ \overline{AB} \perp \overline{BD} \end{array}} & \:\: \textsf{Given} \\ \hline \angle B\textsf{ is a right angle} & 3. \textsf{ Definition of Perpendicularity} \\ \hline \overline{CD} \perp \overline{BD} & 4. \textsf{ Given} \\ \hline \angle D\textsf{ is a right angle} & \textsf{ Definition of Perpendicularity} \\ \hline \angle B \cong \angle D & 5.\textsf{ Right Angle Congruence Theorem} \\ \hline E\textsf{ is the midpoint of }\overline{BD} & \:\:\textsf{Given} \\ \hline \overline{BE}\cong \overline{DE} & 6.\textsf{ Definition of Midpoint} \\ \hline \triangle ABE \cong \triangle CDE & \textsf{SAS Congruence Postulate} \\ \hline \end{array} \end{gathered}STATEMENTSAB≅CDAB⊥BD∠B is a right angleCD⊥BD∠D is a right angle∠B≅∠DE is the midpoint of BDBE≅DE△ABE≅△CDEREASONSGiven3. Definition of Perpendicularity4. Given Definition of Perpendicularity5. Right Angle Congruence TheoremGiven6. Definition of MidpointSAS Congruence Postulate
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