To solve these right-angled triangles, we can use trigonometric ratios (sine, cosine, and tangent) based on the angles provided. Let's go through each triangle.

1. Top Left Triangle:

   • Given: Opposite side = $x$, Hypotenuse = 5 cm, Angle = 29°
   • Using sine: $\sin(29°) = \frac{x}{5}$
   • Solving for $x$: $x = 5 \cdot \sin(29°)$
   • $x \approx 2.4$ cm

2. Top Middle Triangle:

   • Given: Adjacent side = 7 cm, Hypotenuse = $x$, Angle = 17°
   • Using cosine: $\cos(17°) = \frac{7}{x}$
   • Solving for $x$: $x = \frac{7}{\cos(17°)}$
   • $x \approx 7.3$ cm

3. Top Right Triangle:

   • Given: Adjacent side = 4 cm, Opposite side = $x$, Angle = 65°
   • Using tangent: $\tan(65°) = \frac{x}{4}$
   • Solving for $x$: $x = 4 \cdot \tan(65°)$
   • $x \approx 8.6$ cm

4. Bottom Left Triangle:

   • Given: Opposite side = $x$, Hypotenuse = 4.1 cm, Angle = 25°
   • Using sine: $\sin(25°) = \frac{x}{4.1}$
   • Solving for $x$: $x = 4.1 \cdot \sin(25°)$
   • $x \approx 1.7$ cm

5. Bottom Middle Triangle:

   • Given: Opposite side = $x$, Adjacent side = 4.4 cm, Angle = 41°
   • Using tangent: $\tan(41°) = \frac{x}{4.4}$
   • Solving for $x$: $x = 4.4 \cdot \tan(41°)$
   • $x \approx 3.8$ cm

6. Bottom Right Triangle:

   • Given: Adjacent side = 3.1 cm, Opposite side = $x$, Angle = 53°
   • Using tangent: $\tan(53°) = \frac{x}{3.1}$
   • Solving for $x$: $x = 3.1 \cdot \tan(53°)$
   • $x \approx 4.1$ cm

Answers (to 1 decimal place):

1. $x \approx 2.4$ cm
2. $x \approx 7.3$ cm
3. $x \approx 8.6$ cm
4. $x \approx 1.7$ cm
5. $x \approx 3.8$ cm
6. $x \approx 4.1$ cm

Would you like more detailed steps for each calculation, or have any questions?

Related Questions:

1. How do trigonometric functions relate to side lengths in right triangles?
2. What is the difference between using sine, cosine, and tangent in right triangles?
3. How would these calculations change if we knew two side lengths but not the angle?
4. How can we verify these results using the Pythagorean theorem?
5. Can these methods apply to non-right triangles?

Tip:

Always double-check the angle and side placement relative to each other to choose the correct trigonometric ratio!