Question

1. given: ∠1≅∠2, ∠3≅∠4

prove: (overline{ac}congoverline{bd})

Explanation:

Step1: Identify triangles

Consider $\triangle AEC$ and $\triangle BED$.

Step2: Use angle - angle - side (AAS) criterion

In $\triangle AEC$ and $\triangle BED$, we have $\angle 1=\angle 2$ (given), $\angle 3 = \angle 4$ (given), and $BE = CE$ (vertically - opposite angles are equal, and since $\angle 1=\angle 2$ and $\angle 3=\angle 4$, the triangles are congruent by AAS).

Step3: Prove congruence

By the AAS congruence criterion, $\triangle AEC\cong\triangle BED$.

Step4: Use congruent - triangle property

Since $\triangle AEC\cong\triangle BED$, corresponding parts of congruent triangles are congruent. So, $AC = BD$, which means $\overline{AC}\cong\overline{BD}$.

Answer:

$\overline{AC}\cong\overline{BD}$ is proved as shown above.