The diagram shows two parallel lines $n \parallel m$ intersected by a transversal, creating several angles. The key points are:

• $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$ are formed at point $B$.
• There are given angle measures: $\angle A = 20^\circ$ and $\angle C = 60^\circ$.

Let's analyze each option to determine which statements are true:

Step-by-step Analysis

1. Using Angle Relationships

Since $n \parallel m$, we can use the properties of parallel lines cut by a transversal. Here are some key angle relationships:

• Corresponding Angles: Corresponding angles are congruent.
• Alternate Interior Angles: These are equal because the lines are parallel.
• Linear Pair: The sum of two angles on a straight line is $180^\circ$.

2. Calculate $\angle 1$

$\angle A$ and $\angle 1$ are alternate interior angles. Since $\angle A = 20^\circ$, $\angle 1 = 20^\circ$.

3. Calculate $\angle 4$

$\angle C = 60^\circ$ is an exterior angle, and $\angle 4$ is its corresponding angle on the parallel lines. Therefore, $\angle 4 = 60^\circ$.

4. Calculate $\angle 2$ and $\angle 3$

• Since $\angle 1$ and $\angle 2$ form a linear pair (they are on the same straight line at $B$), $\angle 2 = 180^\circ - 20^\circ = 160^\circ$.
• $\angle 3$ is the corresponding angle to $\angle 2$ (alternate interior), so $\angle 3 = 160^\circ$.

Checking Each Statement:

A. $m \angle 2 = 60^\circ$: False. $\angle 2 = 160^\circ$.

B. $m \angle 3 = 100^\circ$: False. $\angle 3 = 160^\circ$.

C. $m \angle 2 + m \angle 4 = 80^\circ$: False. $\angle 2 + \angle 4 = 160^\circ + 60^\circ = 220^\circ$.

D. $m \angle 2 + m \angle 3 = 80^\circ$: False. $\angle 2 + \angle 3 = 160^\circ + 160^\circ = 320^\circ$.

E. $m \angle 2 = 20^\circ$: False. $\angle 2 = 160^\circ$.

Conclusion:

None of the provided options are correct based on the analysis of the angles in the diagram.

Would you like to go over any specific part in more detail or review the logic behind a particular step?

5 Related Questions:

1. What are alternate interior angles, and how are they used to solve angle problems in parallel lines?
2. How do you calculate angles using linear pairs on a straight line?
3. Why do corresponding angles formed by a transversal cutting parallel lines have equal measures?
4. Can two angles add up to more than 180° in such diagrams, and if so, under what conditions?
5. How can we determine the measure of an unknown angle using known angles and angle relationships?

Tip: Remember that when two lines are parallel, alternate interior and corresponding angles are key tools for solving for unknown angles!