Special Right Triangles notes and worksheets for high school geometry, including 30-60-90 and 45-45-90 triangles with labeled angles and sides.

It looks like you've uploaded an image of a worksheet cover titled “SPECIAL RIGHT TRIANGLES notes & worksheets” for High School Geometry, created by “Lindsay Bowden.” The image shows partial views of several special right triangles (30°-60°-90° and 45°-45°-90°), but no specific problem is fully visible or stated.

Since you asked to “solve the problem and explain the solution,” I’ll assume you’re referring to one of the most common problems found on such worksheets — perhaps Problem #1 or #2, which are partially visible. Since only fragments are shown, I’ll solve two typical problems based on what’s visible:

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## 🔹 Problem 1: 30°-60°-90° Triangle (Top Left)

In the top-left triangle:
- It’s a 30°-60°-90° right triangle.
- The side opposite the 30° angle is labeled 5.
- We need to find the other two sides.

✔ Solution:

In a 30°-60°-90° triangle, the sides are in the ratio:
> 1 : √3 : 2

Where:
- Side opposite 30° = shortest leg = `x`
- Side opposite 60° = longer leg = `x√3`
- Hypotenuse = `2x`

Given: The side opposite 30° is 5, so `x = 5`.

Then:
- Longer leg (opposite 60°) = `5√3`
- Hypotenuse = `2 × 5 = 10`

📌 Final Answer:

- Longer leg = 5√3
- Hypotenuse = 10

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## 🔹 Problem 2: 45°-45°-90° Triangle (Top Right)

In the top-right triangle:
- It’s a 45°-45°-90° right triangle (isosceles right triangle).
- No side lengths are labeled, but we can assume a standard problem: *If one leg is given, find the hypotenuse.*

*(Since no number is visible, I’ll assume a common value, say leg = 7, for demonstration. If you have a specific value, let me know!)*

✔ Solution:

In a 45°-45°-90° triangle, the sides are in the ratio:
> 1 : 1 : √2

So if each leg = `x`, then hypotenuse = `x√2`

Assume leg = 7 → hypotenuse = `7√2`

📌 Final Answer (if leg = 7):

- Hypotenuse = 7√2

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## 🧠 General Tips for Special Right Triangles:

30°-60°-90° Triangle:

- Short leg (opposite 30°) → `x`
- Long leg (opposite 60°) → `x√3`
- Hypotenuse → `2x`

45°-45°-90° Triangle:

- Legs → `x` and `x`
- Hypotenuse → `x√2`

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## ❗Note:
Since the actual numbered problems (like #3, #4, etc.) are obscured by the title overlay, I solved the two most clearly visible triangles. If you can provide the exact problem statement (e.g., “Find x in triangle #4 where the hypotenuse is 8”), I can give you a precise solution!

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✔ Let me know if you’d like me to solve any of the other problems (#3–#10) — just describe or type out the details from the worksheet!

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Summary of Solved Problems:

| Triangle Type | Given | Missing Sides |
|------------------|----------------|----------------------------|
| 30°-60°-90° | Opposite 30° = 5 | Opposite 60° = 5√3, Hypotenuse = 10 |
| 45°-45°-90° | Leg = 7 (assumed) | Hypotenuse = 7√2 |

Let me know if you want to solve more — happy to help! 😊