We are given that $ABCD$ is a square and $AE \cong BG$. We need to prove that $\triangle GEF$ is isosceles.

1. Problem Description

We are given that $ABCD$ is a square and $AE \cong BG$. We need to prove that $\triangle GEF$ is isosceles.

2. Solution Steps

Statements | Reasons

------- | --------

1. $ABCD$ is a square | Given

2. $AE \cong BG$ | Given

3. $AB \cong BC$ | Definition of a square (all sides are congruent)

4. $AB = BC$ | Definition of congruence

5. $AB = AE + EB$ and $BC = BG + GC$ | Segment Addition Postulate

6. $AE + EB = BG + GC$ | Substitution Property of Equality (from steps 4 and 5)

7. $AE = BG$ | Definition of congruence (from step 2)

8. $BG + EB = BG + GC$ | Substitution Property of Equality (from steps 6 and 7)

9. $EB = GC$ | Subtraction Property of Equality

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0. $\angle A \cong \angle B$ | Definition of a square (all angles are right angles and congruent)

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1. $\angle A = \angle B$ | Definition of congruence

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2. $\triangle ABE \cong \triangle BCG$ | SAS (Side-Angle-Side) Congruence Theorem (from steps 2, 11 and 9)

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3. $BE = AG$ | Definition of congruence

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4. $BE \cong AG$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

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5. $\angle AEB \cong \angle BGC$ | CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

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6. $AE = EB = AG = BG $ | Side AE and BG congruent and BE = AG. Therefore $AG = AE$.

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7. $EF = EG$ | Triangle EBF and EAG are same and therefore two sides EF and EG are congruent.

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8. $\triangle GEF$ is isosceles | Definition of an isosceles triangle (a triangle with at least two congruent sides)

3. Final Answer

$\triangle GEF$ is isosceles