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The problem provided involves three angles, where:

- ∠1 is supplementary to ∠2
- ∠2 is supplementary to ∠3
- Expressions for each angle are given as follows:
  - ∠1 = 2x + y
  - ∠2 = x + 30
  - ∠3 = 3y + 20

Since the angles are supplementary, this implies that:

- ∠1 + ∠2 = 180° (because they are supplementary)
- ∠2 + ∠3 = 180° (since they are also supplementary)

We can use these relationships to solve for ∠1.

Let’s proceed step by step.

### Step 1: Use the fact that ∠1 + ∠2 = 180°
\[
(2x + y) + (x + 30) = 180
\]
Simplifying:
\[
3x + y + 30 = 180
\]
\[
3x + y = 150 \quad \text{(Equation 1)}
\]

### Step 2: Use the fact that ∠2 + ∠3 = 180°
\[
(x + 30) + (3y + 20) = 180
\]
Simplifying:
\[
x + 3y + 50 = 180
\]
\[
x + 3y = 130 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations
We now have the following system of equations:
1. \( 3x + y = 150 \)
2. \( x + 3y = 130 \)

I will solve this system to find the values of \(x\) and \(y\), and then substitute these values into the expression for ∠1. Let’s proceed with that calculation.The solution to the system of equations is:

- \( x = 40 \)
- \( y = 30 \)

### Step 4: Find ∠1
Now, we substitute these values into the expression for ∠1:

\[
\text{∠1} = 2x + y = 2(40) + 30 = 80 + 30 = 110^\circ
\]

Thus, ∠1 is \( 110^\circ \).

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What would ∠2 be using the values of \(x\) and \(y\)?
2. How would you verify that ∠1 and ∠2 are supplementary?
3. What is the value of ∠3 using the given expressions?
4. What would happen if ∠1 and ∠3 were complementary instead of supplementary?
5. How would the system change if the sum of all three angles was 360°?

**Tip**: Always double-check if the conditions provided, such as angles being supplementary, hold after solving the equations.

Find the value of angle 1 (∠1), given that ∠1 is supplementary to ∠2, and ∠2 is supplementary to ∠3, with the following expressions for the angles: ∠1 = 2x + y, ∠2 = x + 30, and ∠3 = 3y + 20.

This problem involves solving for angle ∠1, using algebraic expressions for three angles that are supplementary. The key concepts include supplementary angles and linear equations. The provided expressions are ∠1 = 2x + y, ∠2 = x + 30, and ∠3 = 3y + 20. Learn how to solve for ∠1 step by step.

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