To solve for the missing angles, let's analyze the given triangle and apply geometric properties such as the sum of angles in a triangle, which always equals $180^\circ$.

Step-by-step Solution:

1. Triangle $ABC$:

   • We are given $\angle BAC = 20^\circ$ and $\angle ABC = 50^\circ$.
   • To find $\angle ACB$: $\angle ACB = 180^\circ - (\angle BAC + \angle ABC)$ $\angle ACB = 180^\circ - (20^\circ + 50^\circ) = 110^\circ$

2. Triangle $ADE$:

   • We know $\angle DAE = 20^\circ$ and $\angle ADE$ is adjacent to the straight line with $\angle BAE$.
   • Since $\angle ABE$ forms a straight line: $\angle ADE = 180^\circ - (\angle BAC + \angle CAE)$ $\angle ADE = 180^\circ - (20^\circ + 30^\circ) = 130^\circ$
   • Now, $\angle AED$ can be found because the sum of angles in triangle $ADE$ is $180^\circ$: $\angle AED = 180^\circ - (\angle DAE + \angle ADE)$ $\angle AED = 180^\circ - (20^\circ + 130^\circ) = 30^\circ$

Summary of Missing Angles:

• $\angle ACB = 110^\circ$
• $\angle ADE = 130^\circ$
• $\angle AED = 30^\circ$

Would you like a more detailed explanation or have any questions about this solution?

Relative Questions to Expand Understanding:

1. How does the sum of angles in a triangle help solve problems like this?
2. What properties of triangles are used to determine missing angles?
3. Can exterior angles of triangles provide an alternative way to solve this problem?
4. How would the problem change if one of the given angles was altered?
5. What role do adjacent angles on a straight line play in finding missing angles?

Tip:

When solving problems involving triangles, always remember that the sum of the interior angles is $180^\circ$. This property is fundamental and useful for determining unknown angles.