The problem asks to prove that triangle $GEF$ is an isosceles triangle, given that $ABCD$ is a square and $AE \cong BG$. We need to provide the statements and reasons in a proof.

1. Problem Description

The problem asks to prove that triangle $GEF$ is an isosceles triangle, given that $ABCD$ is a square and $AE \cong BG$. We need to provide the statements and reasons in a proof.

2. Solution Steps

Statement 1: $ABCD$ is a square, $AE \cong BG$

Reason 1: Given

Statement 2: $AB \cong BC$

Reason 2: Definition of a square (All sides of a square are congruent)

Statement 3: $AB = AE + EB$, $BC = BG + GC$

Reason 3: Segment Addition Postulate

Statement 4: $AE + EB = BG + GC$

Reason 4: Substitution Property (Substitute $AB$ with $AE + EB$ and $BC$ with $BG + GC$ in $AB = BC$)

Statement 5: $AE = BG$

Reason 5: Given (Part of the initial information)

Statement 6: $EB = GC$

Reason 6: Subtraction Property of Equality (Since $AE + EB = BG + GC$ and $AE = BG$, then subtracting $AE$ and $BG$ from both sides gives $EB = GC$)

Statement 7: $\angle B \cong \angle C$

Reason 7: Definition of a square (All angles of a square are right angles, and all right angles are congruent)

Statement 8: $\triangle EBF \cong \triangle GCF$

Reason 8: Angle-Side-Angle (ASA) Congruence Theorem ($\angle B \cong \angle C$, $EB \cong GC$, and $\angle BEF \cong \angle CGF$ because they are vertical angles)

Statement 9: $FE \cong FG$

Reason 9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Since $\triangle EBF \cong \triangle GCF$, $FE \cong FG$

Statement 10: $\triangle GEF$ is isosceles

Reason 10: Definition of Isosceles Triangle (A triangle with two congruent sides is an isosceles triangle)

3. Final Answer

1. $ABCD$ is a square, $AE \cong BG$ ; Given

2. $AB \cong BC$ ; Definition of a square

3. $AB = AE + EB$, $BC = BG + GC$ ; Segment Addition Postulate

4. $AE + EB = BG + GC$ ; Substitution Property

5. $AE = BG$ ; Given

6. $EB = GC$ ; Subtraction Property of Equality

7. $\angle B \cong \angle C$ ; Definition of a square

8. $\triangle EBF \cong \triangle GCF$ ; Angle-Side-Angle (ASA) Congruence Theorem

9. $FE \cong FG$ ; CPCTC

1

0. $\triangle GEF$ is isosceles ; Definition of Isosceles Triangle