Question

complete the flowchart proof. then use this to write a two - column proof.
given: \\( \angle 1 \cong \angle 3 \\)
prove: \\( \angle 2 \cong \angle 4 \\)
\\( \angle 1 \cong \angle 3 \\) \\( \angle 1 \cong \angle 2, \angle 3 \cong \angle 4 \\) \\( \angle 2 \cong \angle 3 \\) \\( \angle 2 \cong \angle 4 \\)
given vertical angles
congruence theorem transitive
property of
angle
congruence transitive
property of
angle
congruence

Explanation:

Two - Column Proof:

Statements	Reasons
\(\angle1\cong\angle2\)	Vertical Angles Congruence Theorem
\(\angle3\cong\angle4\)	Vertical Angles Congruence Theorem
\(\angle2\cong\angle3\)	Transitive Property of Angle Congruence (\(\angle1\cong\angle3\) and \(\angle1\cong\angle2\))
\(\angle2\cong\angle4\)	Transitive Property of Angle Congruence (\(\angle2\cong\angle3\) and \(\angle3\cong\angle4\))

Step1: State the given

We are given that \(\angle1\cong\angle3\).

Step2: Use Vertical Angles Congruence Theorem

By the Vertical Angles Congruence Theorem, \(\angle1\) and \(\angle2\) are vertical angles, so \(\angle1\cong\angle2\). Also, \(\angle3\) and \(\angle4\) are vertical angles, so \(\angle3\cong\angle4\).

Step3: Apply Transitive Property of Angle Congruence for \(\angle2\) and \(\angle3\)

Since \(\angle1\cong\angle3\) (given) and \(\angle1\cong\angle2\) (from Step 2), by the Transitive Property of Angle Congruence (\(If\ a = b\) and \(a = c\), then \(b = c\)), we have \(\angle2\cong\angle3\).

Step4: Apply Transitive Property of Angle Congruence for \(\angle2\) and \(\angle4\)

Since \(\angle2\cong\angle3\) (from Step 3) and \(\angle3\cong\angle4\) (from Step 2), by the Transitive Property of Angle Congruence, we have \(\angle2\cong\angle4\).

Answer:

The two - column proof is shown above, which proves that \(\angle2\cong\angle4\) using the given \(\angle1\cong\angle3\) and the Vertical Angles Congruence Theorem along with the Transitive Property of Angle Congruence.