Question

determine the triangle congruence theorem and right triangle congruence theorem that could be used to prove each statement.
in the figure shown, \\(\overline{ad}\\) and \\(\overline{be}\\) are perpendicular bisectors of each other.
prove \\(\triangle acb \cong \triangle dce\\).
answer attempt 2 out of 2
triangle congruence theorem:
right triangle congruence theorem:

Explanation:

Step1: Identify bisected segments

Since $\overline{AD}$ and $\overline{BE}$ are perpendicular bisectors of each other, $AC = DC$ and $BC = EC$.

Step2: Identify vertical angles

$\angle ACB$ and $\angle DCE$ are vertical angles, so $\angle ACB = \angle DCE$.

Step3: Apply congruence theorem

We have two pairs of equal sides and the included equal angle, so use SAS (Side-Angle-Side).

Step4: Analyze right triangle congruence

The perpendicular bisectors mean $\angle ACB$ and $\angle DCE$ are right angles? No, wait: the bisectors are perpendicular to each other, so $\angle ACB = 90^\circ$. For right triangles, since legs $AC=DC$, $BC=EC$, use SAS (or Leg-Leg, which is a special case of SAS for right triangles, equivalent to SAS).

Answer:

Triangle Congruence Theorem: SAS (Side-Angle-Side)
Right Triangle Congruence Theorem: Leg-Leg (LL, a special case of SAS for right triangles, equivalent to SAS)