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 1 out of 2
triangle congruence theorem: sas
right triangle congruence theorem: hl

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 SAS congruence

We have two pairs of equal sides and the included equal angle: $AC=DC$, $\angle ACB=\angle DCE$, $BC=EC$. By the Side-Angle-Side (SAS) congruence theorem, $\triangle ACB \cong \triangle DCE$.

Step4: Analyze right triangle congruence

Since $\overline{AD} \perp \overline{BE}$, $\triangle ACB$ and $\triangle DCE$ are right triangles with hypotenuses $AB$ and $DE$, and legs $AC=DC$, $BC=EC$. For right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another, we use Hypotenuse-Leg (HL). Here, we can also derive $AB=DE$ from SAS, so HL applies to the right triangles formed by the perpendicular bisector.

Answer:

Triangle Congruence Theorem: SAS
Right Triangle Congruence Theorem: HL