I can help you solve the problems shown in the image. Let’s analyze the questions:

Question 1

In the first figure:

• Given: $\angle 1 = 41^\circ$, $\angle 2 = 25^\circ$, and you need to determine $\angle AED$.

• Explanation:

  • Use the Vertical Angle Theorem, which states that vertical angles are congruent. For example, $\angle 1$ and the angle directly opposite it are equal.
  • Similarly, $\angle AED$ is the sum of $\angle 1$ and $\angle 2$, as it spans those two smaller angles.

  Calculation: $\angle AED = \angle 1 + \angle 2 = 41^\circ + 25^\circ = 66^\circ$

Question 2

In the second figure:

• Given: $\angle 1 = 2x + 24^\circ$, $\angle 2 = 3x$. You need to determine the value of $x$ and then find each angle measure.

• Explanation:

  • Angles on a straight line add up to $180^\circ$. Thus: $\angle 1 + \angle 2 = 180^\circ$
  • Substitute the given values: $(2x + 24) + 3x = 180$
  • Solve for $x$: $5x + 24 = 180$ $5x = 156$ $x = 31.2$
  • Use $x$ to calculate the angles: $\angle 1 = 2(31.2) + 24 = 62.4 + 24 = 86.4^\circ$ $\angle 2 = 3(31.2) = 93.6^\circ$

  The measures of $\angle 1$ and $\angle 2$ are $86.4^\circ$ and $93.6^\circ$, respectively.

Would you like a detailed explanation of these calculations or additional help with similar problems? Here are some related questions you might consider:

1. How is the Vertical Angle Theorem applied in different configurations?
2. Why do angles on a straight line sum to $180^\circ$?
3. What are alternate interior angles, and when are they equal?
4. How do you determine if two lines are parallel using angle relationships?
5. How can you check if your angle calculations are correct?

Tip: Always double-check your angle sums to ensure they satisfy geometric properties, such as $180^\circ$ for a straight line or $360^\circ$ for a full circle.