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Triangles review answers
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GPT-4o mini
Sure! If you have specific questions or problems related to triangles that you need help with, feel free to share them, and I can provide you with explanations or answers. Here are some general concepts related to triangles that might help you in your review:
1. **Types of Triangles**:
- **By Sides**:
- Equilateral: All sides are equal.
- Isosceles: Two sides are equal.
- Scalene: All sides are different.
- **By Angles**:
- Acute: All angles are less than 90 degrees.
- Right: One angle is exactly 90 degrees.
- Obtuse: One angle is greater than 90 degrees.
2. **Triangle Sum Theorem**: The sum of the interior angles of a triangle is always 180 degrees.
3. **Pythagorean Theorem**: In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): \( c^2 = a^2 + b^2 \).
4. **Area of a Triangle**: The area (A) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
or using Heron's formula if all three sides (a, b, c) are known:
\[ s = \frac{a+b+c}{2} \]
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
5. **Congruence Criteria** for triangles:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- HL (Hypotenuse-Leg for right triangles)
6. **Similarity Criteria**: Triangles are similar if:
- AA (Angle-Angle)
- SSS (Side-Side-Side proportional)
- SAS (Side-Angle-Side proportional)
Feel free to ask if you want to dive deeper into any of these topics or if you have specific problems to tackle!
1. **Types of Triangles**:
- **By Sides**:
- Equilateral: All sides are equal.
- Isosceles: Two sides are equal.
- Scalene: All sides are different.
- **By Angles**:
- Acute: All angles are less than 90 degrees.
- Right: One angle is exactly 90 degrees.
- Obtuse: One angle is greater than 90 degrees.
2. **Triangle Sum Theorem**: The sum of the interior angles of a triangle is always 180 degrees.
3. **Pythagorean Theorem**: In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): \( c^2 = a^2 + b^2 \).
4. **Area of a Triangle**: The area (A) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
or using Heron's formula if all three sides (a, b, c) are known:
\[ s = \frac{a+b+c}{2} \]
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
5. **Congruence Criteria** for triangles:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- HL (Hypotenuse-Leg for right triangles)
6. **Similarity Criteria**: Triangles are similar if:
- AA (Angle-Angle)
- SSS (Side-Side-Side proportional)
- SAS (Side-Angle-Side proportional)
Feel free to ask if you want to dive deeper into any of these topics or if you have specific problems to tackle!