https://file.aiquickdraw.com/mathbot/file/1726713322429jux90rqx.jpg

The problem states that:

Given:
- \( m\angle 1 = 2x + 40 \)
- \( m\angle 2 = 2y + 40 \)
- \( m\angle 3 = x + 2y \)

We are asked to find \( m\angle 1 \), \( m\angle 2 \), and \( m\angle 3 \).

### Step-by-step solution:

1. From the diagram, we see that angles \( \angle 1 \) and \( \angle 2 \) are vertical angles, meaning they are congruent. Therefore, we can set:
   \[
   2x + 40 = 2y + 40
   \]
   Simplifying this equation:
   \[
   2x = 2y
   \]
   \[
   x = y
   \]

2. Now substitute \( x = y \) into the equation for \( m\angle 3 \):
   \[
   m\angle 3 = x + 2y = y + 2y = 3y
   \]

3. Since \( \angle 1 \) and \( \angle 3 \) are supplementary (they form a linear pair), their sum is 180°:
   \[
   m\angle 1 + m\angle 3 = 180
   \]
   Substitute the expressions for \( m\angle 1 \) and \( m\angle 3 \):
   \[
   (2y + 40) + 3y = 180
   \]
   Simplify:
   \[
   5y + 40 = 180
   \]
   \[
   5y = 140
   \]
   \[
   y = 28
   \]

4. Now, substitute \( y = 28 \) back into the expressions for the angles:
   - \( m\angle 1 = 2y + 40 = 2(28) + 40 = 96^\circ \)
   - \( m\angle 2 = 2y + 40 = 2(28) + 40 = 96^\circ \)
   - \( m\angle 3 = 3y = 3(28) = 84^\circ \)

### Final answers:
- \( m\angle 1 = 96^\circ \)
- \( m\angle 2 = 96^\circ \)
- \( m\angle 3 = 84^\circ \)

Would you like more details or have any other questions?

### Related Questions:
1. How can you determine when two angles are vertical?
2. What is the property of linear pairs of angles?
3. How do you verify if two angles are supplementary?
4. How do you solve for variables when dealing with multiple equations?
5. Can you apply these techniques to more complex geometric problems?

**Tip:** Always look for relationships like vertical angles, linear pairs, or congruent angles in geometric diagrams for quick solutions.

Given: m∠1 = 2x + 40, m∠2 = 2y + 40, m∠3 = x + 2y. Find: m∠1, m∠2, and m∠3.

Learn how to solve for the measures of angles in a geometric diagram where vertical and supplementary angles are involved. This problem combines algebraic equations and key angle theorems, making it suitable for Grades 8-10.

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