We are given two triangle problems. 1) In triangle $ABC$, $C = 96.2^\circ$, $b = 11.2$ cm, and $c = 39.4$ cm. We need to solve the triangle completely. 2) In triangle $XYZ$, $Y = 29.8^\circ$, $Z = 51.4^\circ$, and $x = 19.6$ cm. We need to solve the triangle completely.

1. Problem Description

We are given two triangle problems.

1) In triangle $ABC$, $C = 96.2^\circ$, $b = 11.2$ cm, and $c = 39.4$ cm. We need to solve the triangle completely.

2) In triangle $XYZ$, $Y = 29.8^\circ$, $Z = 51.4^\circ$, and $x = 19.6$ cm. We need to solve the triangle completely.

2. Solution Steps

Problem 1: Triangle $ABC$

We are given $C$, $b$, and $c$.

We can use the Law of Sines to find angle $B$.

$\frac{\sin B}{b} = \frac{\sin C}{c}$

$\sin B = \frac{b \sin C}{c}$

$\sin B = \frac{11.2 \sin(96.2^\circ)}{39.4}$

$\sin B = \frac{11.2 \times 0.9940}{39.4} \approx \frac{11.1328}{39.4} \approx 0.2825$

$B = \arcsin(0.2825) \approx 16.39^\circ$

Now we can find angle $A$:

$A = 180^\circ - B - C$

$A = 180^\circ - 16.39^\circ - 96.2^\circ$

$A = 180^\circ - 112.59^\circ = 67.41^\circ$

Now we can find side $a$:

Using Law of Sines:

$\frac{a}{\sin A} = \frac{c}{\sin C}$

$a = \frac{c \sin A}{\sin C}$

$a = \frac{39.4 \sin(67.41^\circ)}{\sin(96.2^\circ)}$

$a = \frac{39.4 \times 0.9234}{0.9940}$

$a = \frac{36.382}{0.9940} \approx 36.60$ cm

Problem 2: Triangle $XYZ$

We are given $Y$, $Z$, and $x$.

We can find angle $X$ first:

$X = 180^\circ - Y - Z$

$X = 180^\circ - 29.8^\circ - 51.4^\circ$

$X = 180^\circ - 81.2^\circ = 98.8^\circ$

Now we can find side $y$ using Law of Sines:

$\frac{y}{\sin Y} = \frac{x}{\sin X}$

$y = \frac{x \sin Y}{\sin X}$

$y = \frac{19.6 \sin(29.8^\circ)}{\sin(98.8^\circ)}$

$y = \frac{19.6 \times 0.4972}{0.9881}$

$y = \frac{9.74512}{0.9881} \approx 9.86$ cm

Now we can find side $z$ using Law of Sines:

$\frac{z}{\sin Z} = \frac{x}{\sin X}$

$z = \frac{x \sin Z}{\sin X}$

$z = \frac{19.6 \sin(51.4^\circ)}{\sin(98.8^\circ)}$

$z = \frac{19.6 \times 0.7820}{0.9881}$

$z = \frac{15.3272}{0.9881} \approx 15.51$ cm

3. Final Answer

Problem 1:

$A = 67.41^\circ$

$B = 16.39^\circ$

$a = 36.60$ cm

Problem 2:

$X = 98.8^\circ$

$y = 9.86$ cm

$z = 15.51$ cm