Question

take note: angle theorems

• alternate interior angles ________

if two parallel lines are cut by a transversal, the pairs of alternate interior angles are congruent.

• alternate exterior angles ________

if two parallel lines are cut by a transversal, the pairs of alternate exterior angles are congruent.

• corresponding angles ________

if two parallel lines are cut by a transversal, the pairs of corresponding angles are congruent.

• consecutive (same-side) interior angles theorem ________

if two parallel lines are cut by a transversal, the pairs of consecutive interior angles are supplementary.

• consecutive (same-side) exterior angles theorem ________

if two parallel lines are cut by a transversal, the pairs of consecutive exterior angles are

Explanation:

To fill in the blanks for the angle theorems, we recall the definitions of each type of angle formed by parallel lines and a transversal:

Alternate Interior Angles

These are angles that lie between the two parallel lines (interior) and on opposite sides of the transversal. For example, in the diagram, ∠4 and ∠5, or ∠3 and ∠6 are alternate interior angles. The theorem states they are congruent when lines are parallel.

Alternate Exterior Angles

These lie outside the two parallel lines (exterior) and on opposite sides of the transversal. For example, ∠1 and ∠8, or ∠2 and ∠7 are alternate exterior angles. The theorem states they are congruent when lines are parallel.

Corresponding Angles

These are angles that occupy the same relative position at each intersection where a straight line crosses two others. For example, ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are corresponding angles. The theorem states they are congruent when lines are parallel.

Consecutive (Same - Side) Interior Angles Theorem

These lie between the two parallel lines (interior) and on the same side of the transversal. For example, ∠3 and ∠5, or ∠4 and ∠6 are consecutive interior angles. The theorem states they are supplementary (sum to \(180^\circ\)) when lines are parallel.

Consecutive (Same - Side) Exterior Angles Theorem

These lie outside the two parallel lines (exterior) and on the same side of the transversal. For example, ∠1 and ∠7, or ∠2 and ∠8 are consecutive exterior angles. The theorem states they are supplementary (sum to \(180^\circ\)) when lines are parallel.

If we assume the blanks are for the definitions (what the angles are), the answers would be:

• Alternate Interior Angles: Angles between the parallel lines, on opposite sides of the transversal.
• Alternate Exterior Angles: Angles outside the parallel lines, on opposite sides of the transversal.
• Corresponding Angles: Angles in the same relative position at each intersection.
• Consecutive (Same - Side) Interior Angles Theorem: Angles between the parallel lines, on the same side of the transversal (and supplementary).
• Consecutive (Same - Side) Exterior Angles Theorem: Angles outside the parallel lines, on the same side of the transversal (and supplementary).

If the blanks are for the "name" of the relationship (e.g., "are congruent" for some, "are supplementary" for others), we can use the following:

• Alternate Interior Angles: are congruent
• Alternate Exterior Angles: are congruent
• Corresponding Angles: are congruent
• Consecutive (Same - Side) Interior Angles Theorem: are supplementary
• Consecutive (Same - Side) Exterior Angles Theorem: are supplementary

Final Answers (for the relationship blanks, as per typical angle theorem notes):

• Alternate Interior Angles: \(\boldsymbol{\text{are congruent}}\)
• Alternate Exterior Angles: \(\boldsymbol{\text{are congruent}}\)
• Corresponding Angles: \(\boldsymbol{\text{are congruent}}\)
• Consecutive (Same - Side) Interior Angles Theorem: \(\boldsymbol{\text{are supplementary}}\)
• Consecutive (Same - Side) Exterior Angles Theorem: \(\boldsymbol{\text{are supplementary}}\)

Answer:

To fill in the blanks for the angle theorems, we recall the definitions of each type of angle formed by parallel lines and a transversal:

Alternate Interior Angles

These are angles that lie between the two parallel lines (interior) and on opposite sides of the transversal. For example, in the diagram, ∠4 and ∠5, or ∠3 and ∠6 are alternate interior angles. The theorem states they are congruent when lines are parallel.

Alternate Exterior Angles

These lie outside the two parallel lines (exterior) and on opposite sides of the transversal. For example, ∠1 and ∠8, or ∠2 and ∠7 are alternate exterior angles. The theorem states they are congruent when lines are parallel.

Corresponding Angles

These are angles that occupy the same relative position at each intersection where a straight line crosses two others. For example, ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are corresponding angles. The theorem states they are congruent when lines are parallel.

Consecutive (Same - Side) Interior Angles Theorem

These lie between the two parallel lines (interior) and on the same side of the transversal. For example, ∠3 and ∠5, or ∠4 and ∠6 are consecutive interior angles. The theorem states they are supplementary (sum to \(180^\circ\)) when lines are parallel.

Consecutive (Same - Side) Exterior Angles Theorem

These lie outside the two parallel lines (exterior) and on the same side of the transversal. For example, ∠1 and ∠7, or ∠2 and ∠8 are consecutive exterior angles. The theorem states they are supplementary (sum to \(180^\circ\)) when lines are parallel.

If we assume the blanks are for the definitions (what the angles are), the answers would be:

• Alternate Interior Angles: Angles between the parallel lines, on opposite sides of the transversal.
• Alternate Exterior Angles: Angles outside the parallel lines, on opposite sides of the transversal.
• Corresponding Angles: Angles in the same relative position at each intersection.
• Consecutive (Same - Side) Interior Angles Theorem: Angles between the parallel lines, on the same side of the transversal (and supplementary).
• Consecutive (Same - Side) Exterior Angles Theorem: Angles outside the parallel lines, on the same side of the transversal (and supplementary).

If the blanks are for the "name" of the relationship (e.g., "are congruent" for some, "are supplementary" for others), we can use the following:

• Alternate Interior Angles: are congruent
• Alternate Exterior Angles: are congruent
• Corresponding Angles: are congruent
• Consecutive (Same - Side) Interior Angles Theorem: are supplementary
• Consecutive (Same - Side) Exterior Angles Theorem: are supplementary

Final Answers (for the relationship blanks, as per typical angle theorem notes):

• Alternate Interior Angles: \(\boldsymbol{\text{are congruent}}\)
• Alternate Exterior Angles: \(\boldsymbol{\text{are congruent}}\)
• Corresponding Angles: \(\boldsymbol{\text{are congruent}}\)
• Consecutive (Same - Side) Interior Angles Theorem: \(\boldsymbol{\text{are supplementary}}\)
• Consecutive (Same - Side) Exterior Angles Theorem: \(\boldsymbol{\text{are supplementary}}\)