Question

1. proof copy and complete the flowchart proof. then write a two - column proof. (see example 1) given ∠1≅∠3 prove ∠2≅∠4 ∠1≅∠3→∠1≅∠2,∠3≅∠4→∠2≅∠3→∠2≅∠4 given vertical angles congruence theorem (theorem 2.6)

Explanation:

Step1: Recall vertical - angles property

$\angle1\cong\angle2$ and $\angle3\cong\angle4$ because vertical angles are congruent.

Step2: Use the given information

We are given that $\angle1\cong\angle3$.

Step3: Apply the transitive property of congruence

Since $\angle1\cong\angle2$, $\angle3\cong\angle4$ and $\angle1\cong\angle3$, we can conclude that $\angle2\cong\angle4$ (if $a = b$, $b = c$, then $a = c$ for congruent angles).

Answer:

The missing reasons in the flowchart proof are: Transitive Property of Congruence (for the third box) and Transitive Property of Congruence (for the fourth box).

Two - column proof:

Statements	Reasons
2. $\angle1\cong\angle2$	Vertical Angles Congruence Theorem
3. $\angle3\cong\angle4$	Vertical Angles Congruence Theorem
4. $\angle2\cong\angle3$	Transitive Property of Congruence (from 1 and 2)
5. $\angle2\cong\angle4$	Transitive Property of Congruence (from 3 and 4)