Question

decide whether enough information is given to prove $\triangle pqt \cong \triangle srt$ using the hl congruence theorem. explain.
$\angle ptq$ and $\square$ are right angles,
$\overline{pt} \cong \square$ and $\overline{qt} \cong \square$ so, the triangles $\square$ be proven congruent using the hl congruence theorem because $\square$

Explanation:

1. First, identify the right angles in the diagram: the marked right angles are $\angle PTQ$ and $\angle STR$.
2. Next, match the congruent segments marked on the diagram: $\overline{PT} \cong \overline{ST}$, and $\overline{QT} \cong \overline{RT}$.
3. The HL Congruence Theorem requires a right angle, a congruent hypotenuse, and a congruent leg for each right triangle. Here, $\triangle PQT$ and $\triangle SRT$ are right triangles, we have congruent legs ($\overline{PT} \cong \overline{ST}$), but the congruent segments $\overline{QT}$ and $\overline{RT}$ are the legs, not the hypotenuses (the hypotenuses are $\overline{PQ}$ and $\overline{SR}$, which are not marked as congruent). So HL cannot be used.

Answer:

$\angle PTQ$ and $\boldsymbol{\angle STR}$ are right angles,
$\overline{PT} \cong \boldsymbol{\overline{ST}}$ and $\overline{QT} \cong \boldsymbol{\overline{RT}}$
So, the triangles $\boldsymbol{cannot}$ be proven congruent using the HL Congruence Theorem because
we only have congruent corresponding legs, not a congruent hypotenuse and leg for each triangle