1. **State the problem:** We want to find values of $x$ and $y$ such that two right triangles are congruent by the HL (Hypotenuse-Leg) theorem. 2. **Recall the HL theorem:** Two right triangles are congruent if their hypotenuses are equal and one corresponding leg is equal. 3. **Problem 5:** - Triangle 1 sides: $x$, $x+3$, right angle. - Triangle 2 sides: $3y$, $y+1$, right angle. 4. **Identify hypotenuses:** The hypotenuse is the longest side. - For Triangle 1, hypotenuse is $x+3$ (since $x+3 > x$ for positive $x$). - For Triangle 2, hypotenuse is $3y$ or $y+1$ depending on values, but generally $3y$ is larger if $y>0$. 5. **Set hypotenuses equal:** $$x+3 = 3y$$ 6. **Set one leg equal:** Choose the shorter leg for congruence: $$x = y+1$$ 7. **Solve the system:** From $x = y+1$, substitute into $x+3=3y$: $$ (y+1) + 3 = 3y $$ $$ y + 4 = 3y $$ $$ 4 = 3y - y $$ $$ 4 = 2y $$ $$ y = 2 $$ Then, $$ x = y + 1 = 2 + 1 = 3 $$ 8. **Check:** - Hypotenuses: $x+3 = 3 + 3 = 6$, $3y = 3 \times 2 = 6$ equal. - Legs: $x = 3$, $y+1 = 2 + 1 = 3$ equal. **Answer for problem 5:** $x=3$, $y=2$. --- Since the user asked multiple problems but per instructions we solve only the first, we stop here.