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We are tasked with finding all the angles of the triangle with the given vertices \( K(3, 4) \), \( L(-2, 3) \), and \( M(4, 3) \).

### Step 1: Calculate the lengths of the sides of the triangle
We will use the distance formula to find the lengths of each side:
\[
\text{Distance between two points } (x_1, y_1) \text{ and } (x_2, y_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Side KL:
\[
KL = \sqrt{((-2) - 3)^2 + (3 - 4)^2} = \sqrt{(-5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26}
\]

#### Side LM:
\[
LM = \sqrt{(4 - (-2))^2 + (3 - 3)^2} = \sqrt{(6)^2 + 0^2} = \sqrt{36} = 6
\]

#### Side KM:
\[
KM = \sqrt{(4 - 3)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

### Step 2: Use the Law of Cosines to find the angles
The Law of Cosines states:
\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Where \( \theta \) is the angle opposite the side \( c \), and \( a \) and \( b \) are the lengths of the other two sides.

#### Angle \( \angle K \) (opposite side LM):
\[
\cos(K) = \frac{KL^2 + KM^2 - LM^2}{2 \cdot KL \cdot KM}
\]
Substitute the values:
\[
\cos(K) = \frac{(\sqrt{26})^2 + (\sqrt{2})^2 - 6^2}{2 \cdot \sqrt{26} \cdot \sqrt{2}} = \frac{26 + 2 - 36}{2 \cdot \sqrt{52}} = \frac{-8}{2 \cdot \sqrt{52}} = \frac{-8}{14.422}
\]
\[
K = \cos^{-1}\left( \frac{-8}{14.422} \right)
\]

#### Angle \( \angle L \) (opposite side KM):
Use the Law of Cosines again for angle \( L \).

I will continue solving and calculating all angles next.The angles of the triangle with vertices \( K(3, 4) \), \( L(-2, 3) \), and \( M(4, 3) \) are:

- \( \angle K \approx 123.69^\circ \)
- \( \angle L \approx 11.31^\circ \)
- \( \angle M = 45.0^\circ \)

Would you like further details or have any questions about the process?

Here are 5 related questions to deepen understanding:
1. How can you use the Law of Sines to check the angles of a triangle?
2. What is the significance of the sum of angles in a triangle?
3. How would the angles change if one of the vertices was moved?
4. How can you classify this triangle based on its angles?
5. What role do coordinate geometry principles play in solving geometric problems?

**Tip:** Always ensure that the sum of angles in a triangle equals \(180^\circ\) to verify your results.

Find all angles of the triangle with vertices K(3, 4), L(-2, 3), and M(4, 3).

Solve for all the angles of a triangle given vertices at K(3, 4), L(-2, 3), and M(4, 3). This problem applies the distance formula and the Law of Cosines, making it ideal for high school students in grades 9-12.

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