This question has two parts. First, answer Part A. Then, answer Part B. Part A A triangle has sides with lengths 8,15 , and 17 . Which of the following verifies this is a Pythagorean triple? O A) $$ 8^{2}+15^{2}=17^{2} $$ B) $$ 8^{2}+17^{2}=15^{2} $$ C) $$ 15^{2}+17^{2}=8^{2} $$ Part B Approximate these side lengths do not form a Pythagorean triple. ascending order. D angles in this triangle to the nearest degree and list in Degrees and

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**Part A**

To verify whether the side lengths $$8$$, $$15$$, and $$17$$ form a Pythagorean triple, we check if the Pythagorean theorem holds true. The theorem states that in a right-angled triangle:

$$
\text{(Leg 1)}^{2} + \text{(Leg 2)}^{2} = \text{(Hypotenuse)}^{2}
$$

Let's evaluate each option:

- **Option A:** $$8^{2} + 15^{2} = 17^{2}$$
  $$
  64 + 225 = 289 \quad \Rightarrow \quad 289 = 289 \quad \text{(True)}
  $$
  
- **Option B:** $$8^{2} + 17^{2} = 15^{2}$$
  $$
  64 + 289 = 225 \quad \Rightarrow \quad 353 \neq 225 \quad \text{(False)}
  $$
  
- **Option C:** $$15^{2} + 17^{2} = 8^{2}$$
  $$
  225 + 289 = 64 \quad \Rightarrow \quad 514 \neq 64 \quad \text{(False)}
  $$

**Answer:** **Option A) $$8^{2} + 15^{2} = 17^{2}$$** verifies that the side lengths form a Pythagorean triple.

---

**Part B**

Given that the triangle with sides $$8$$, $$15$$, and $$17$$ is a right-angled triangle (as verified in Part A), we can find the measures of the three angles.

1. **Identify the Right Angle:**
   - The longest side, $$17$$, is the hypotenuse.
   - Therefore, the angle opposite to the hypotenuse is $$90^\circ$$.

2. **Calculate the Other Two Angles:**
   - **Angle opposite the side of length $$8$$:**
     $$
     \sin(\theta) = \frac{8}{17} \quad \Rightarrow \quad \theta = \sin^{-1}\left(\frac{8}{17}\right) \approx 28^\circ
     $$
     
   - **Angle opposite the side of length $$15$$:**
     $$
     \sin(\phi) = \frac{15}{17} \quad \Rightarrow \quad \phi = \sin^{-1}\left(\frac{15}{17}\right) \approx 62^\circ
     $$

3. **Arrange the Angles in Ascending Order:**
   $$
   28^\circ, \quad 62^\circ, \quad 90^\circ
   $$

**Answer:** The triangle's angles, in ascending order and rounded to the nearest degree, are **28°, 62°, and 90°**.
**Part A:** Option A) $$8^{2} + 15^{2} = 17^{2}$$ verifies the Pythagorean triple. **Part B:** The triangle's angles, in ascending order, are 28°, 62°, and 90°.

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