1. **Problem 1:** Find the missing angle measures $\angle C$, $\angle A$, and side $a$ in triangle $ABC$ where $AB=20$ cm, $BC=19$ cm, and $\angle A=65^\circ$. Also find $\angle F$, $\angle D$, and side $d$ in triangle $DEF$ with $DE=19$ cm, $EF=d$, $DF=20$ cm, and $\angle F=65^\circ$ (obtuse). Assume $\angle C$ is acute and $\angle F$ is obtuse. 2. **Step 1:** Use the Law of Cosines to find side $a$ in triangle $ABC$: $$a^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle A)$$ $$a^2 = 20^2 + 19^2 - 2 \cdot 20 \cdot 19 \cdot \cos(65^\circ)$$ Calculate: $$a^2 = 400 + 361 - 760 \cdot 0.4226 = 761 - 321.976 = 439.024$$ $$a = \sqrt{439.024} \approx 20.95 \text{ cm}$$ Rounded to nearest cm: $a = 21$ cm. 3. **Step 2:** Find $\angle C$ using Law of Sines: $$\frac{a}{\sin(\angle A)} = \frac{BC}{\sin(\angle C)}$$ $$\sin(\angle C) = \frac{BC \cdot \sin(\angle A)}{a} = \frac{19 \cdot \sin(65^\circ)}{20.95}$$ $$\sin(\angle C) = \frac{19 \cdot 0.9063}{20.95} = \frac{17.219}{20.95} = 0.822$$ $$\angle C = \arcsin(0.822) \approx 55.1^\circ$$ Rounded: $\angle C = 55^\circ$. 4. **Step 3:** Find $\angle B$: $$\angle B = 180^\circ - \angle A - \angle C = 180^\circ - 65^\circ - 55^\circ = 60^\circ$$ 5. **Step 4:** For triangle $DEF$, $\angle F$ is obtuse and $\angle F = 65^\circ$ is given, so the obtuse angle is $180^\circ - 65^\circ = 115^\circ$. 6. **Step 5:** Use Law of Cosines to find side $d$ opposite $\angle D$: $$d^2 = DE^2 + DF^2 - 2 \cdot DE \cdot DF \cdot \cos(\angle F)$$ $$d^2 = 19^2 + 20^2 - 2 \cdot 19 \cdot 20 \cdot \cos(115^\circ)$$ Calculate: $$d^2 = 361 + 400 - 760 \cdot (-0.4226) = 761 + 321.976 = 1082.976$$ $$d = \sqrt{1082.976} \approx 32.91 \text{ cm}$$ Rounded: $d = 33$ cm. 7. **Step 6:** Find $\angle D$ using Law of Sines: $$\frac{d}{\sin(\angle F)} = \frac{DE}{\sin(\angle D)}$$ $$\sin(\angle D) = \frac{DE \cdot \sin(\angle F)}{d} = \frac{19 \cdot \sin(115^\circ)}{32.91}$$ $$\sin(\angle D) = \frac{19 \cdot 0.9063}{32.91} = \frac{17.219}{32.91} = 0.523$$ $$\angle D = \arcsin(0.523) \approx 31.5^\circ$$ Rounded: $\angle D = 32^\circ$. 8. **Step 7:** Find $\angle E$: $$\angle E = 180^\circ - \angle D - \angle F = 180^\circ - 32^\circ - 115^\circ = 33^\circ$$ --- 9. **Problem 2:** Given triangle with $\angle B = 10^\circ$, sides $AB=15$ cm, $BC=15$ cm, and a smaller triangle with side $EF=5$ cm, find missing angles $\angle C$, $\angle A$, and side $a$ in the first triangle, and $\angle F$, $\angle D$, and side $d$ in the second. 10. **Step 1:** Since $AB=BC=15$ cm and $\angle B=10^\circ$, triangle $ABC$ is isosceles with base $AC$. 11. **Step 2:** Find $\angle A$ and $\angle C$: $$\angle A = \angle C = \frac{180^\circ - 10^\circ}{2} = 85^\circ$$ 12. **Step 3:** Use Law of Sines to find side $a = AC$: $$\frac{a}{\sin(10^\circ)} = \frac{15}{\sin(85^\circ)}$$ $$a = \frac{15 \cdot \sin(10^\circ)}{\sin(85^\circ)} = \frac{15 \cdot 0.1736}{0.9962} = 2.61 \text{ cm}$$ 13. **Step 4:** For the smaller triangle with side $EF=5$ cm, and right or obtuse known angle, both other angles are acute. 14. **Step 5:** Since the problem does not provide explicit angles or sides for the second triangle, we cannot calculate $\angle F$, $\angle D$, or $d$ without additional information. --- **Final answers:** **Problem 1:** - $m\angle C = 55^\circ$ - $m\angle A = 65^\circ$ - $a = 21$ cm - $m\angle F = 115^\circ$ - $m\angle D = 32^\circ$ - $d = 33$ cm **Problem 2:** - $m\angle C = 85^\circ$ - $m\angle A = 85^\circ$ - $a = 2.61$ cm - $m\angle F$, $m\angle D$, and $d$ cannot be determined with given data.