Question

you try:
given: ∠1 is supplementary to ∠2, ∠2 is supplementary to ∠3, ∠3 is supplementary to ∠4, and ∠4 is supplementary to ∠1
prove: ∠2 ≅ ∠4
hint: all of the reasons from above will stay the same!!

Explanation:

Step1: Recall Supplementary Angles

Supplementary angles sum to \(180^\circ\). Given \(\angle1\) is supplementary to \(\angle2\), so \(m\angle1 + m\angle2 = 180^\circ\). Also, \(\angle2\) is supplementary to \(\angle3\), so \(m\angle2 + m\angle3 = 180^\circ\). And \(\angle3\) is supplementary to \(\angle4\), so \(m\angle3 + m\angle4 = 180^\circ\), and \(\angle4\) is supplementary to \(\angle1\), so \(m\angle4 + m\angle1 = 180^\circ\).

Step2: Use Congruence of Supplements

From \(m\angle1 + m\angle2 = 180^\circ\) and \(m\angle4 + m\angle1 = 180^\circ\), by the Congruent Supplements Theorem (if two angles are supplementary to the same angle, they are congruent), \(\angle2 \cong \angle4\). Also, from \(m\angle2 + m\angle3 = 180^\circ\) and \(m\angle3 + m\angle4 = 180^\circ\), \(\angle2 \cong \angle4\) (same theorem).

Now, constructing the proof table:

Statements	Reasons
2. \(\angle4\) is supplementary to \(\angle1\)	Given
3. \(\angle2 \cong \angle4\)	Congruent Supplements Theorem (if two angles are supplementary to the same angle, they are congruent)

Answer:

Using the Congruent Supplements Theorem, we prove \(\angle2 \cong \angle4\) as shown in the proof table. The key is recognizing that \(\angle2\) and \(\angle4\) are both supplementary to \(\angle1\) (or \(\angle3\)), so they must be congruent.