https://file.aiquickdraw.com/mathbot/file/1726885265649qihe40sq.jpg

In the figure, you're given the following information:
- \( \angle 4 = 90^\circ \)
- \( \angle 5 = 50^\circ \)
- \( \angle 6 = 40^\circ \)

The task is to find the measures of \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \).

### Step-by-step solution:

1. **Use vertical angles:**
   - \( \angle 1 \) and \( \angle 5 \) are vertical angles, so they must have the same measure:
     \[
     \angle 1 = \angle 5 = 50^\circ
     \]

2. **Use linear pairs:**
   - \( \angle 4 \) and \( \angle 2 \) form a linear pair, which means they add up to \( 180^\circ \):
     \[
     \angle 4 + \angle 2 = 180^\circ
     \]
     Since \( \angle 4 = 90^\circ \), we can solve for \( \angle 2 \):
     \[
     90^\circ + \angle 2 = 180^\circ
     \]
     \[
     \angle 2 = 180^\circ - 90^\circ = 90^\circ
     \]

3. **Use another linear pair:**
   - \( \angle 3 \) and \( \angle 6 \) form a linear pair, so:
     \[
     \angle 3 + \angle 6 = 180^\circ
     \]
     Given \( \angle 6 = 40^\circ \), we can solve for \( \angle 3 \):
     \[
     \angle 3 = 180^\circ - 40^\circ = 140^\circ
     \]

### Final answers:
- \( \angle 1 = 50^\circ \)
- \( \angle 2 = 90^\circ \)
- \( \angle 3 = 140^\circ \)

Would you like further details on any of these steps? Here are 5 related questions to explore further:

1. How do vertical angles help solve problems involving intersecting lines?
2. What is the linear pair theorem and how is it applied here?
3. Why do angles in a linear pair always add up to \(180^\circ\)?
4. How can you identify other angle relationships in this diagram?
5. What are some examples where vertical angles and linear pairs are used in real-world applications?

**Tip:** In any geometry problem involving intersecting lines, always look for vertical angles and linear pairs—they can help simplify the solution quickly!

In the figure below, the measure of ∠4 = 90°, the measure of ∠5 = 50°, and the measure of ∠6 = 40°. What are the measures of ∠1, ∠2, and ∠3?

Learn how to find the measures of angles using the Vertical Angles Theorem and Linear Pair Theorem. This problem involves calculating the angles of ∠1, ∠2, and ∠3, given the values of ∠4, ∠5, and ∠6. Ideal for students in Grades 8-10.

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