Question

given: \\( \angle 3 \cong \angle 4 \\)
prove: \\( \angle 1 \cong \angle 2 \\)

1 \\( \angle 3 \cong \angle 4 \\) given

2 \\( \angle \square \\) and \\( \angle \square \\) form a linear pair
\\( \angle \square \\) and \\( \angle \square \\) form a linear pair definition of linear pair

3 \\( \angle \square \\) and \\( \angle \square \\) are supplementary
\\( \angle \square \\) and \\( \angle \square \\) are supplementary select

4 \\( \angle \square \cong \angle \square \\) select

Explanation:

Step1: Identify linear pairs

From the diagram, $\angle1$ and $\angle3$ form a linear pair, $\angle2$ and $\angle4$ form a linear pair.

Step2: Use linear pair property

If two angles form a linear pair, they are supplementary. So $\angle1$ and $\angle3$ are supplementary ($\angle1+\angle3 = 180^{\circ}$), $\angle2$ and $\angle4$ are supplementary ($\angle2+\angle4=180^{\circ}$).

Step3: Apply congruent - supplements theorem

If two angles are congruent ($\angle3\cong\angle4$ given) and their supplements exist, then the supplements of the congruent angles are congruent.

Answer:

1. $\angle1$ and $\angle3$; $\angle2$ and $\angle4$
2. $\angle1$ and $\angle3$; $\angle2$ and $\angle4$ (Reason: If two angles form a linear pair, then they are supplementary)
3. $\angle1\cong\angle2$ (Reason: Congruent - Supplements Theorem)