Question

there is a diagram of two parallel lines cut by a transversal, forming angles 1,2,3,4,5,6,7,8. the options are:
□ ∠1 and ∠7 are alternate interior angles.
□ ∠2 and ∠8 are alternate interior angles.
□ ∠3 and ∠5 are alternate exterior angles.
□ ∠4 and ∠7 are same - side exterior angles.
□ ∠5 and ∠1 are same - side interior angles.
□ ∠6 and ∠2 are same - side interior angles.

Explanation:

To solve this, we analyze each pair using angle - related concepts (alternate interior, alternate exterior, same - side interior, same - side exterior angles) when two parallel lines are cut by a transversal:

1. Analyze ∠1 and ∠7

For alternate interior angles, they should lie between the two parallel lines and on opposite sides of the transversal. ∠1 is above the upper parallel line, and ∠7 is below the lower parallel line. So, ∠1 and ∠7 are not alternate interior angles.

2. Analyze ∠2 and ∠8

Alternate interior angles lie between the two parallel lines and on opposite sides of the transversal. ∠2 and ∠8 are between the two parallel lines and on opposite sides of the transversal. So, ∠2 and ∠8 are alternate interior angles.

3. Analyze ∠3 and ∠5

Alternate exterior angles lie outside the two parallel lines and on opposite sides of the transversal. ∠3 is above the upper parallel line, and ∠5 is below the lower parallel line. So, ∠3 and ∠5 are not alternate exterior angles.

4. Analyze ∠4 and ∠7

Same - side exterior angles lie outside the two parallel lines and on the same side of the transversal. ∠4 and ∠7 do not meet this criterion. So, ∠4 and ∠7 are not same - side exterior angles.

5. Analyze ∠5 and ∠1

Same - side interior angles lie between the two parallel lines and on the same side of the transversal. ∠5 and ∠1 do not meet this criterion. So, ∠5 and ∠1 are not same - side interior angles.

6. Analyze ∠6 and ∠2

Same - side interior angles lie between the two parallel lines and on the same side of the transversal. ∠6 and ∠2 do not meet this criterion. So, ∠6 and ∠2 are not same - side interior angles.

We check each angle pair against the definitions of alternate interior, alternate exterior, same - side interior, and same - side exterior angles. Only ∠2 and ∠8 satisfy the definition of alternate interior angles.

Answer:

$\boldsymbol{\angle 2}$ and $\boldsymbol{\angle 8}$ are alternate interior angles (the option "∠2 and ∠8 are alternate interior angles" should be selected).