The red angles, ∠C & ∠E, are on one side of the transversal, in between the two black lines. Therefore, ∠C & ∠E are consecutive interior angles. The blue angles, ∠D & ∠F, are also on one side of the transversal (in this case, on the other side), in between the two lines. Therefore, ∠D & ∠F are consecutive interior angles. ∠A, ∠B, ∠G, and ∠H are not in between the two lines. Therefore, they are not consecutive interior angles.

We see two parallel lines crossed by a transversal. Angles of the same value have matching colors. The purple angles = 150°, while the pink angles = 30°. Look closely at the two consecutive interior angles on the left side of the transversal. The top one (in pink) is 30°, while the bottom one (in purple) is 150°. By adding them, we see that 30°+ 150° = 180°. Likewise, the consecutive interior angles on the right side of the transversal are 150° and 30°, which add up to 180°.

How do we prove that each pair of consecutive interior angles adds up to 180°, without knowing what those angles are?

This means that ∠A = ∠D, and ∠B = ∠C. Likewise, ∠E = ∠H, and ∠F = ∠G.

∠A = ∠E ∠B = ∠F ∠C = ∠G ∠D = ∠H

Since ∠A + ∠B = 180°, ∠D + ∠F = 180°. Since ∠G + ∠H = 180°, ∠C + ∠E = 180° as well. This proves that both pairs of consecutive interior angles add up to 180°.

Since 45° + 135° = 180°, both pairs of consecutive interior angles add up to 180°. We know that consecutive interior angles only add up to 180° when the transversal intersects parallel lines. Therefore, since our consecutive interior angles add up to 180°, we know that the lines must be parallel.

How can we prove that the lines are parallel, without knowing the exact values of the angles?

We know that consecutive interior angles only add up to 180° when a transversal crosses parallel lines. Therefore we can deduce that the lines in the example above must be parallel.

We know that ∠A + ∠B = 180°, since they come together to form a line. The same is true for ∠C + ∠D, ∠E + ∠F, and ∠G + ∠H. [9] X Research source We also know that ∠A = ∠D, ∠B = ∠C, ∠E = ∠H, and ∠F = ∠G, since intersecting lines produce identical angles on opposite sides. Lastly, we know that the angles on the top line are replicated on the bottom line, since a transversal always intersects parallel lines at the same angles. Therefore ∠A = ∠E, ∠B = ∠F, ∠C = ∠G, and ∠D = ∠H. Since ∠A = ∠E and ∠B = ∠C, we know that ∠A + ∠B = ∠E + ∠C. Since ∠A + ∠B = 180°, ∠E + ∠C = 180° as well. We can use the same process to show that ∠D + ∠F = 180°. Therefore, the consecutive interior angles are supplementary.

∠1 + ∠3 = 180°, since they come together to form a line. ∠2 + ∠4, ∠5 + ∠7, and ∠6 + ∠8 = 180° as well. [10] X Research source We know that ∠1 = ∠4, ∠2 = ∠3, ∠5 = ∠8, and ∠6 = ∠7, since intersecting lines produce identical angles on opposite sides. We also know that the angles on the left line are replicated on the right line, since a transversal always intersects parallel lines at the same angles. Thus ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8. Since ∠1 = ∠5 and ∠2 = ∠3, we know that ∠1 + ∠3 = ∠5 + ∠2. Since ∠1 + ∠3 = 180°, ∠5 + ∠2 = 180°. The same process shows that ∠4 + ∠7 = 180°. Therefore, the consecutive interior angles are supplementary.

The consecutive interior angles are ∠Y & ∠A and ∠Z & ∠B. ∠W + ∠X = 180°, since they come together to form a line. ∠Y + ∠Z, ∠A + ∠B, and ∠C + ∠D = 180° as well. Since intersecting lines produce identical angles on opposite sides, we know that ∠W = ∠Z, ∠X = ∠Y, ∠A = ∠D, and ∠B = ∠C. We also know that the angles on the top line are replicated on the bottom line, since a transversal always intersects parallel lines at the same angles. Thus ∠W = ∠A, ∠X = ∠B, ∠Y = ∠C, and ∠Z = ∠D. Since ∠W = ∠Z and ∠X = ∠B, we know that ∠W + ∠X = ∠Z + ∠B. Since ∠W + ∠X = 180°, ∠Z + ∠B = 180°. The same process shows that ∠Y + ∠A = 180°. Therefore, the consecutive interior angles are supplementary.