Question

given: $\triangle jkl$ and $\triangle lmj$ are right triangles, $\overline{jk} \cong \overline{lm}$
prove: $\angle jlk \cong \angle ljm$
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$$\begin{tabular}{|c|c|} \\hline statements & reasons \\\\ \\hline 1. $\\triangle jkl$ and $\\triangle lmj$ are right triangles & 1. \\\\ \\hline 2. & 2. given \\\\ \\hline 3. $\\overline{jl} \\cong \\overline{lj}$ & 3. \\\\ \\hline 4. & 4. hypotenuse - leg (hl) \\\\ \\hline 5. $\\angle jlk \\cong \\angle ljm$ & 5. \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: State given right triangles

Given: $\triangle JKL$ and $\triangle LMJ$ are right triangles

Step2: State given congruent legs

$\overline{JK} \cong \overline{LM}$

Step3: Identify common hypotenuse

$\overline{JL} \cong \overline{LJ}$ (Reflexive Property of Congruence: a segment is congruent to itself)

Step4: Apply HL congruence

$\triangle JKL \cong \triangle LMJ$ (Hypotenuse-Leg (HL) congruence for right triangles: if hypotenuse and one leg of a right triangle are congruent to hypotenuse and one leg of another right triangle, the triangles are congruent)

Step5: Corresponding angles congruent

$\angle JLK \cong \angle LJM$ (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Answer:

1. Given
2. $\overline{JK} \cong \overline{LM}$
3. Reflexive Property of Congruence
4. $\triangle JKL \cong \triangle LMJ$
5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)