Question: User: 5.4 Equilateral and Isosceles Triangles Dave Jean-Baptiste EQUILATERAL TRIANGLES A triangle is equilateral if and only if it is equiangular If $m\angle A = m\angle B = m\angle C$, then $AB = BC = AC$ If $AB = BC = AC$, then $m\angle A = m\angle B = m\angle C = 60^\circ$ Directions: Find each missing measure. 11. $m\angle C = 60^\circ$ $BC = 9$ ft $AC = 9$ ft 12. $m\angle J =$ $m\angle K =$ $m\angle L =$ $24$ cm 13. If $\triangle PQR$ is an equilateral triangle, solve for $x$ and $y$.

1. **Problem 11:** Given $m\angle C = 60^\circ$, $BC = 9$ ft, and $AC = 9$ ft, find missing measures. Since $BC = AC$, triangle $ABC$ is isosceles with $AB$ unknown. 2. **Equilateral triangle property:** If all sides are equal, all angles are $60^\circ$. 3. Given $BC = AC = 9$ ft, if $AB$ is also $9$ ft, triangle $ABC$ is equilateral. 4. Since $m\angle C = 60^\circ$ and $BC = AC$, $m\angle A = m\angle B = 60^\circ$. 5. Therefore, $AB = 9$ ft and $m\angle A = m\angle B = 60^\circ$. --- 6. **Problem 12:** Given triangle $JKL$ with base $24$ cm and missing angles $m\angle J$, $m\angle K$, $m\angle L$. 7. If triangle $JKL$ is equilateral, all sides equal $24$ cm. 8. Therefore, all angles are equal: $m\angle J = m\angle K = m\angle L = 60^\circ$. --- 9. **Problem 13:** $\triangle PQR$ is equilateral; solve for $x$ and $y$. 10. Since $\triangle PQR$ is equilateral, all sides equal and all angles $60^\circ$. 11. Use given expressions for sides or angles involving $x$ and $y$ (not provided in prompt). 12. Set equal sides equal and solve system of equations for $x$ and $y$. 13. Without explicit expressions, cannot solve numerically here. **Final answers:** - Problem 11: $AB = 9$ ft, $m\angle A = m\angle B = 60^\circ$ - Problem 12: $m\angle J = m\angle K = m\angle L = 60^\circ$ - Problem 13: Requires expressions to solve for $x$ and $y$.