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solve this using the law of sines (answer: <C=114°)

This is a triangle where we need to solve for angle \( \angle C \) using the **Law of Sines**. The given values are:

- \( \angle A = 46^\circ \)
- \( \angle B = 20^\circ \)
- Side \( AB = 65 \) (opposite \( C \))

### Step 1: Use the angle sum property of a triangle
We know the sum of the angles in any triangle is \( 180^\circ \). Thus, we can find \( \angle C \) by:

\[
\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 46^\circ - 20^\circ = 114^\circ
\]

So, \( \angle C = 114^\circ \).

### Step 2: Apply the Law of Sines
The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Here:
- \( a \) is the side opposite \( \angle A \)
- \( b \) is the side opposite \( \angle B \)
- \( c \) is the side opposite \( \angle C \) (which is given as 65)

Since \( \angle C = 114^\circ \), the given problem is already solved for \( \angle C \).

Let me know if you'd like to find the missing sides using the Law of Sines, or if you have any further questions!

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**Relative Questions:**
1. How do you apply the Law of Cosines for solving triangles?
2. What is the relationship between the Law of Sines and Law of Cosines?
3. How would you solve a triangle if all sides were known but no angles were given?
4. How can we determine if a triangle is acute, obtuse, or right using its angles?
5. What happens if two angles and a non-included side are given in a triangle?

**Tip:** Always check that the sum of the angles in a triangle equals \( 180^\circ \) to ensure your calculations are consistent!

Solve this triangle using the law of sines (answer: angle C = 114°).

Learn how to solve triangle ABC using the Law of Sines. Given that angle A = 46°, angle B = 20°, and side AB = 65, find the missing angle C and check the steps to ensure the angles sum to 180°. Suitable for students in Grades 9-12.

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