Let's analyze each of the statements given in the problem to determine if they are true or false based on the properties of the angles shown in the diagram.

Given Diagram:

• The diagram shows two parallel lines cut by a transversal.

Analysis of Each Statement:

A. $\angle 3 \cong \angle 2$ because they are alternate interior angles.

• False. $\angle 3$ and $\angle 2$ are on the same side of the transversal and are not alternate interior angles. Alternate interior angles would be on opposite sides of the transversal.

B. $m\angle 1 + m\angle 3 = 180$ because they form a straight angle.

• True. $\angle 1$ and $\angle 3$ are supplementary because they form a linear pair, which sums up to 180 degrees.

C. $\angle 3 \cong \angle 6$ because they are alternate interior angles.

• True. $\angle 3$ and $\angle 6$ are on opposite sides of the transversal and inside the two parallel lines, making them alternate interior angles.

D. $\angle 1$ and $\angle 6$ are supplementary because $\angle 3 \cong \angle 6$ and $m\angle 1 + m\angle 3 = 180$.

• False. While $\angle 3 \cong \angle 6$, $\angle 1$ and $\angle 6$ are not supplementary based on this relationship. $\angle 1$ and $\angle 6$ are corresponding angles, and corresponding angles are congruent, not supplementary.

E. $\angle 1 \cong \angle 3$ because they are vertical angles.

• False. $\angle 1$ and $\angle 3$ are not vertical angles. Vertical angles are formed when two lines intersect, and they are opposite each other.

True Statements:

• B and C are the true statements.

Would you like more details on any of these explanations or have any further questions?

Related Questions:

1. What are the properties of alternate interior angles?
2. How can you identify corresponding angles in a transversal setup?
3. What is the difference between supplementary and complementary angles?
4. How do you determine if two angles are vertical angles?
5. What are the angle relationships when parallel lines are cut by a transversal?

Tip:

Remember that alternate interior angles are congruent when two parallel lines are cut by a transversal. They lie on opposite sides of the transversal but inside the parallel lines.