Question: The HL Congruence Theorem Practice Complete the statement to correctly prove $H \cong K$ when each corresponding pair of angles and one side are congruent. Use the image to answer the question. $\triangle ABC$ and $\triangle PQR$ are similar in ratio 1 to 1 length. $AB$ and $PQ$. Suppose that $\angle B$ and $\angle Q$ are right angles, and $\angle A \cong \angle P$, and side $AB$ is congruent by the HL Congruence Theorem. Assume that the legs only are the transverse rotations of $\angle C$ and $\angle R$. What is the length of the side of $\triangle ABC$? $AB = \square \quad BC = \square \quad PQ = \square$

1. **State the problem:** We are given two right triangles $\triangle ABC$ and $\triangle PQR$ with $\angle B$ and $\angle Q$ right angles, $\angle A \cong \angle P$, and side $AB \cong PQ$. We want to find the lengths of sides $AB$, $BC$, and $PQ$ in $\triangle ABC$. 2. **Recall the HL Congruence Theorem:** This theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. 3. **Given information:** - $\angle B$ and $\angle Q$ are right angles. - $\angle A \cong \angle P$. - $AB \cong PQ$ (one leg congruent). - The triangles are similar with ratio 1:1, so corresponding sides are equal. 4. **Use the right triangle properties:** Since $\triangle ABC$ and $\triangle PQR$ are congruent by HL, their corresponding sides are equal. 5. **Assign lengths:** - Let $AB = x$ (vertical leg). - Let $BC = y$ (horizontal leg). - Since $PQ = AB$, $PQ = x$. 6. **Use the Pythagorean theorem:** For $\triangle ABC$, $$AC^2 = AB^2 + BC^2 = x^2 + y^2$$ 7. **Since the triangles are congruent and similar with ratio 1:1, corresponding hypotenuses are equal:** $$AC = PR$$ 8. **From the image and problem, if $AB$ and $PQ$ are given or can be measured, substitute values.** 9. **Final answer:** - $AB = x$ - $BC = y$ - $PQ = x$ Without specific numeric values from the image, the lengths remain as variables $x$ and $y$ with $PQ = AB = x$.