Thank you for the correction. Let's revisit the problem with the correct information:

Given:

• $m\angle 1 = 80^\circ$
• $m\angle 2 = 90^\circ$
• $m\angle 3$, $m\angle 4$, $m\angle 5$, and $m\angle 6$ are the angles we need to determine.

Step 1: Identify Relationships Between the Angles

From the diagram and given angles:

• $m\angle 2 = 90^\circ$ implies that $\overline{AB}$ is perpendicular to $\overline{AC}$, forming a right angle at $\angle ABC$.
• $\overline{AB}$ and $\overline{AC}$ form triangle $ABC$ with $\angle BAC$ being $\angle 1$ and the total sum of angles in any triangle being $180^\circ$.

Step 2: Calculate the Other Angles

Using the properties of triangles and supplementary angles:

1. Find $m\angle 3$:

   • $\angle 3$ is supplementary to $\angle 2$ because they form a straight line. Therefore: $m\angle 3 = 180^\circ - m\angle 2 = 180^\circ - 90^\circ = 90^\circ$

2. Find $m\angle 4$:

   • $\angle 4$ and $\angle 1$ are angles on the same line, thus they sum to 180°: $m\angle 4 = 180^\circ - m\angle 1 = 180^\circ - 80^\circ = 100^\circ$

3. Find $m\angle 5$:

   • Since $\angle 5$ is part of the triangle $ABC$, and we already know $\angle 2$ and $\angle 1$, the sum of the angles in the triangle $ABC$ is 180°: $m\angle 5 = 180^\circ - (m\angle 1 + m\angle 2) = 180^\circ - (80^\circ + 90^\circ) = 180^\circ - 170^\circ = 10^\circ$

4. Find $m\angle 6$:

   • If $m\angle 6$ is an external angle (outside triangle), you would use the external angle theorem: $m\angle 6 = m\angle 1 + m\angle 3 = 80^\circ + 90^\circ = 170^\circ$

Final Answer:

• $m\angle 1 = 80^\circ$
• $m\angle 2 = 90^\circ$
• $m\angle 3 = 90^\circ$
• $m\angle 4 = 100^\circ$
• $m\angle 5 = 10^\circ$
• $m\angle 6 = 170^\circ$

Would you like to go through any particular angle calculation in more detail, or do you have any questions?

Here are 5 related questions for further practice:

1. How would the solution change if $m\angle 1$ were $70^\circ$ instead of $80^\circ$?
2. What are the properties of an isosceles triangle, and how can they help in finding unknown angles?
3. How do supplementary and complementary angles differ in their properties?
4. What is the exterior angle theorem, and how can it be applied to more complex figures?
5. Can you determine the type of triangle formed by $\angle 1 = 50^\circ$, $\angle 2 = 60^\circ$, and $\angle 3 = 70^\circ$?

Tip: Remember that the sum of the angles in any triangle is always $180^\circ$. This fact is often the key to solving many geometric problems involving triangles.