Let's solve the problem step by step.

We are given:

• $m\angle 1 = 2x + 40$
• $m\angle 2 = 2y + 40$
• $m\angle 3 = x + 2y$

From the diagram, angles $\angle 1$ and $\angle 2$ appear to be vertically opposite angles, which means they are congruent (equal). Therefore, we can set their measures equal to each other: $2x + 40 = 2y + 40$ By subtracting 40 from both sides: $2x = 2y$ Dividing both sides by 2: $x = y$

Now that we know $x = y$, we can substitute this into the equation for $m\angle 3$: $m\angle 3 = x + 2y = y + 2y = 3y$

Next, we know that the sum of the angles on a straight line is 180°. Thus: $m\angle 1 + m\angle 3 = 180^\circ$ Substituting $m\angle 1 = 2x + 40$ and $m\angle 3 = 3y$, and knowing $x = y$: $(2y + 40) + 3y = 180$ Simplifying the equation: $5y + 40 = 180$ Subtracting 40 from both sides: $5y = 140$ Dividing by 5: $y = 28$

Now that we have $y = 28$, and $x = y$, so $x = 28$, we can find the measures of the angles:

• $m\angle 1 = 2x + 40 = 2(28) + 40 = 56 + 40 = 96^\circ$
• $m\angle 2 = 2y + 40 = 2(28) + 40 = 96^\circ$
• $m\angle 3 = 3y = 3(28) = 84^\circ$

Thus, the measures of the angles are:

• $m\angle 1 = 96^\circ$
• $m\angle 2 = 96^\circ$
• $m\angle 3 = 84^\circ$

Would you like more details on any step or further explanations?

Here are five related questions to explore:

1. How do you identify vertically opposite angles in any geometric figure?
2. Can you solve a problem where angles are given in terms of different variables?
3. How does the concept of supplementary angles help in solving linear pairs of angles?
4. What are other ways to verify that $\angle 1$ and $\angle 2$ are equal in the diagram?
5. What happens if we change the expressions for the angles? How does the solution process adjust?

Tip: In geometry, identifying angle relationships such as vertical angles, corresponding angles, and supplementary angles is key to solving many problems involving intersecting lines.