Question

if yes, select the appropriate statements and pairs of congruent corresponding parts to show that the triangles are congruent using the hypotenuse - leg triangle congruency theorem. if no, select cannot be proved.
\\( \overline { b d } \cong \overline { b d } \\)
\\( \overline { b c } \cong \overline { a d } \\)
\\( \overline { a b } \cong \overline { c d } \\)
\\( \overline { a b } \cong \overline { b c } \\)
\\( \triangle a b d \\) and \\( \triangle c b d \\) are right triangles.
\\( \triangle a b d \\) and \\( \triangle a b c \\) are right triangles.
cannot be proved.
your answer is partially correct.

Explanation:

Step1: Identify the right - angled triangles

For the Hypotenuse - Leg (HL) Congruency Theorem, we need two right - angled triangles. $\triangle ABD$ and $\triangle CBD$ are right - triangles (this is a correct statement for applying HL). $\triangle ABD$ and $\triangle ABC$ are not the two triangles we should consider for the HL theorem as we need two triangles where we can identify hypotenuse and leg.

Step2: Identify the hypotenuse and leg

The hypotenuse - leg theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.
In right - triangles $\triangle ABD$ and $\triangle CBD$, $\overline{BD}$ is the common hypotenuse ($\overline{BD}\cong\overline{BD}$ by the reflexive property of congruence). If we assume that $\overline{BC}\cong\overline{AD}$ (leg) and $\overline{AB}\cong\overline{CD}$ (hypotenuse is wrong, it should be that if we use HL for $\triangle ABD$ and $\triangle CBD$, one is hypotenuse and one is leg. Wait, no: for right - triangles $\triangle ABD$ and $\triangle CBD$, if we consider $\overline{BD}$ as the hypotenuse (common), and say $\overline{AB}$ and $\overline{CD}$ as legs (if $\overline{AB}\cong\overline{CD}$) or $\overline{AD}$ and $\overline{BC}$ as legs. But actually, for HL, in $\triangle ABD$ and $\triangle CBD$, $\overline{BD}$ is the hypotenuse (common), and if we have $\overline{AD}\cong\overline{BC}$ (legs) and they are right - triangles, then by HL $\triangle ABD\cong\triangle CBD$. The statement $\triangle ABD$ and $\triangle CBD$ are right - triangles is correct. The statement $\overline{BD}\cong\overline{BD}$ (reflexive property) and $\overline{AD}\cong\overline{BC}$ (leg) are the correct parts for HL. The statement $\overline{AB}\cong\overline{CD}$ is not relevant for HL (if we are using HL for $\triangle ABD$ and $\triangle CBD$) and $\triangle ABD$ and $\triangle ABC$ being right - triangles is wrong for the pair of triangles we need (HL needs two right - triangles, and the correct pair is $\triangle ABD$ and $\triangle CBD$).

Answer:

$\overline{BD}\cong\overline{BD}$, $\overline{BC}\cong\overline{AD}$, $\triangle ABD$ and $\triangle CBD$ are right - triangles.