https://file.aiquickdraw.com/mathbot/file/1725420831853rk90ko94.jpg

The image shows two right-angled triangles with some given angles and side lengths.

### Analyzing the Triangles:

1. **Left Triangle:**
   - It is a right-angled triangle.
   - Given angles: \(45^\circ\) and \(45^\circ\), indicating this is an isosceles right triangle.
   - Given side: One side (the leg opposite one of the \(45^\circ\) angles) is marked as 4 units.

2. **Right Triangle:**
   - This is also a right-angled triangle.
   - Given angles: \(30^\circ\), \(60^\circ\), and the right angle \(90^\circ\).
   - Given side: The side opposite the \(30^\circ\) angle is marked as 6 units.

### Solving for the Unknown Sides:

Let's solve for the missing sides in both triangles:

1. **Left Triangle:**
   - Since it is a 45-45-90 triangle, both legs are equal. So, if one leg is 4, the other leg is also 4.
   - In a 45-45-90 triangle, the hypotenuse is \(\sqrt{2}\) times the length of a leg.
   \[
   \text{Hypotenuse} = 4 \cdot \sqrt{2} \approx 5.66
   \]

2. **Right Triangle:**
   - In a 30-60-90 triangle, the ratio of the sides opposite the \(30^\circ\), \(60^\circ\), and \(90^\circ\) angles is \(1 : \sqrt{3} : 2\).
   - Given that the side opposite the \(30^\circ\) angle is 6, the hypotenuse (opposite the \(90^\circ\) angle) will be:
   \[
   \text{Hypotenuse} = 2 \times 6 = 12
   \]
   - The side opposite the \(60^\circ\) angle will be:
   \[
   \text{Side opposite } 60^\circ = 6 \cdot \sqrt{3} \approx 10.39
   \]

Would you like to know more details or ask any questions? 

### Related Questions:
1. How do you derive the side ratios for 30-60-90 and 45-45-90 triangles?
2. What are the properties of isosceles right triangles?
3. How can trigonometry be used to find unknown sides in right triangles?
4. What are some real-life applications of special right triangles?
5. How do you find the area of these triangles?

### Tip:
When working with right triangles, remember that the Pythagorean theorem and trigonometric ratios can help solve for unknown sides or angles efficiently!

Solve the right-angled triangles with the given angles and sides.

Explore how to solve right-angled triangles with given angles and side lengths. This problem covers key concepts in geometry and trigonometry, including the properties of 45-45-90 and 30-60-90 triangles. Suitable for students in Grades 8-10.

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