Let's go through each problem step-by-step to find the required values for $x$. I'll calculate the unknown sides and angles using trigonometric rules, mainly the Sine Rule and the Cosine Rule.

First Image

1. Top Left Triangle:

• Given:

  • $\angle 84^\circ$ and $\angle 32^\circ$ in a triangle.
  • Side opposite $\angle 32^\circ$ is $5.8$ cm.

• Find $x$ (side opposite $\angle 84^\circ$).

Using the Sine Rule: $\frac{x}{\sin(84^\circ)} = \frac{5.8}{\sin(32^\circ)}$ Solving for $x$: $x = \frac{5.8 \times \sin(84^\circ)}{\sin(32^\circ)}$

2. Top Right Triangle:

• Given:

  • $\angle 80^\circ$ and $\angle 67^\circ$.
  • Side opposite $\angle 67^\circ$ is $7.4$ cm.

• Find $x$ (side opposite $\angle 80^\circ$).

Using the Sine Rule: $\frac{x}{\sin(80^\circ)} = \frac{7.4}{\sin(67^\circ)}$ Solving for $x$: $x = \frac{7.4 \times \sin(80^\circ)}{\sin(67^\circ)}$

Second Image

3. Bottom Left Triangle:

• Given:

  • Side lengths $6$ cm and $4.6$ cm.
  • Angle $44^\circ$.

• Find $x$ (unknown angle).

Using the Cosine Rule to find $x$: [ x = \cos^{-1}\left(\frac{6^2 + 4.6^2 - c^2}{2 \times 6 \times 4.6