Question

given, ∠1 ≅ ∠2, transitive property of angle congruence, ∠2 ≅ ∠3, ∠1 ≅ ∠3, angle addition postulate, vertical angles congruence theorem

Explanation:

Step1: Identify Given Statements

We are given $\angle 1 \cong \angle 2$ and $\angle 2 \cong \angle 3$. Also, we know the Transitive Property of Angle Congruence which states that if $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Step2: Apply Transitive Property

Using the Transitive Property of Angle Congruence with $\angle 1 \cong \angle 2$ (first given) and $\angle 2 \cong \angle 3$ (second given), we can conclude that $\angle 1 \cong \angle 3$.

Answer:

To prove $\angle 1 \cong \angle 3$:

1. Given: $\angle 1 \cong \angle 2$ and $\angle 2 \cong \angle 3$.
2. Transitive Property of Angle Congruence: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$. Applying this property with $A = 1$, $B = 2$, and $C = 3$, we get $\angle 1 \cong \angle 3$.

So the final conclusion is $\boldsymbol{\angle 1 \cong \angle 3}$ by the Transitive Property of Angle Congruence.