Question

visit www.bigideasmathvideos.com to watch the flipped video instruction for the \try this\ prob try this video for extra example 5 for 5b - using angle relationships 5) use the two - column proof to write a paragraph proof. given ∠1 ≅ ∠4 prove ∠2 ≅ ∠3 two - column proof statements 1. ∠1 ≅ ∠4 2. ∠1 ≅ ∠2 3. ∠2 ≅ ∠4 4. ∠3 ≅ ∠4 5. ∠2 ≅ ∠3 reasons 1. given 2. vertical angles theorem 3. transitive property of angle congruence 4. vertical angles theorem 5. transitive property of angle congruence

Explanation:

We start with the given that $\angle 1 \cong \angle 4$. By the Vertical Angles Theorem, $\angle 1 \cong \angle 2$ (since they are vertical angles) and $\angle 3 \cong \angle 4$ (since they are vertical angles). Using the Transitive Property of Angle Congruence, since $\angle 1 \cong \angle 4$ and $\angle 1 \cong \angle 2$, we can conclude $\angle 2 \cong \angle 4$. Then, again using the Transitive Property of Angle Congruence, since $\angle 2 \cong \angle 4$ and $\angle 3 \cong \angle 4$, we can conclude $\angle 2 \cong \angle 3$.

Answer:

Paragraph Proof:
We are given that $\angle 1 \cong \angle 4$. By the Vertical Angles Theorem, vertical angles are congruent, so $\angle 1 \cong \angle 2$ and $\angle 3 \cong \angle 4$. Applying the Transitive Property of Angle Congruence to $\angle 1 \cong \angle 4$ and $\angle 1 \cong \angle 2$, we find $\angle 2 \cong \angle 4$. Using the Transitive Property of Angle Congruence once more with $\angle 2 \cong \angle 4$ and $\angle 3 \cong \angle 4$, we can prove that $\angle 2 \cong \angle 3$.