Question

which of the following statements are true? select all that apply. diagram of two parallel lines cut by a transversal, with angles labeled 1,2,3,4 (top intersection) and 5,6,7,8 (bottom intersection) a. ∠3 ≅ ∠2 because they are alternate interior angles. b. ( mangle 1 + mangle 3 = 180^circ ) because they form a straight angle. c. ∠3 ≅ ∠6 because they are alternate interior angles.

Explanation:

Step 1: Analyze Option A

Alternate interior angles are formed when a transversal crosses two parallel lines, and they lie between the two lines on opposite sides of the transversal. $\angle 3$ and $\angle 2$ are adjacent angles forming a linear pair, not alternate interior angles. So, Option A is false.

Step 2: Analyze Option B

A straight angle measures $180^\circ$. $\angle 1$ and $\angle 3$ are adjacent angles that form a straight line (linear pair), so their measures should add up to $180^\circ$. Wait, the option says $110^\circ$? Wait, no, probably a typo, but the concept: linear pair angles sum to $180^\circ$. Wait, the original option: " $m\angle 1 + m\angle 3 = 180^\circ$ because they form a straight angle" (assuming the $110^\circ$ is a typo, or maybe I misread). Wait, looking at the diagram, $\angle 1$ and $\angle 3$ are adjacent and form a linear pair, so their sum is $180^\circ$. So if the option has $180^\circ$ (maybe a typo in the user's image, but the logic: linear pair sums to $180^\circ$). So Option B: if it's $180^\circ$, then it's true. Wait, the user's image: " $m\angle 1 + m\angle 3 = 180^\circ$ because they form a straight angle" (maybe the $110$ is a typo). So assuming that, Option B is true.

Step 3: Analyze Option C

Alternate interior angles: when a transversal crosses two parallel lines, alternate interior angles are equal. $\angle 3$ and $\angle 6$: the two horizontal lines are parallel (assuming, as they are cut by a transversal), $\angle 3$ and $\angle 6$ lie between the two parallel lines, on opposite sides of the transversal. So they are alternate interior angles, hence $\angle 3 \cong \angle 6$. So Option C is true.

Answer:

B. $m\angle 1 + m\angle 3 = 180^\circ$ because they form a straight angle, C. $\angle 3 \cong \angle 6$ because they are alternate interior angles