**Proof: Bisectors of Interior Angles on the Same Sides of a Transversal**

To prove that the bisectors of the interior angles on the same sides of a transversal intersect at right angles, we will use the properties of parallel lines and angles formed by a transversal.

**Given:**
- Two parallel lines, line AB and line CD.
- A transversal line EF intersecting lines AB and CD at points E and F, respectively.

**To prove:**
- The bisectors of the interior angles on the same sides of the transversal EF intersect at right angles.

**Proof:**

1. Let's assume that line AB and line CD are parallel.
2. We have the transversal line EF intersecting line AB and line CD at points E and F, respectively.
3. Four interior angles are formed on the same sides of the transversal EF. Let's label them as ∠1, ∠2, ∠3, and ∠4.
4. We will prove that the bisectors of ∠1 and ∠3 intersect at right angles, and the bisectors of ∠2 and ∠4 intersect at right angles.
- Bisector of ∠1: Let's name it as line l1.
- Bisector of ∠3: Let's name it as line l2.
- Bisector of ∠2: Let's name it as line l3.
- Bisector of ∠4: Let's name it as line l4.
5. To prove that l1 and l2 intersect at right angles, we need to show that the angles formed by l1 and l2 are complementary.
6. Since l1 is the bisector of ∠1, it divides ∠1 into two congruent angles, let's name them as ∠A1E and ∠E1B.
7. Similarly, l2 is the bisector of ∠3, so it divides ∠3 into two congruent angles, let's name them as ∠C1F and ∠F1D.
8. From the properties of parallel lines, we know that ∠A1E and ∠C1F are congruent, and ∠E1B and ∠F1D are congruent.
9. Therefore, we have ∠A1E ≅ ∠C1F and ∠E1B ≅ ∠F1D.
10. Since congruent angles are equal in measure, we can say that ∠A1E = ∠C1F and ∠E1B = ∠F1D.
11. By the definition of complementary angles, two angles are complementary if their sum is equal to 90 degrees.
12. We can see that ∠A1E + ∠E1B = ∠C1F + ∠F1D = 90 degrees.
13. Thus, the bisectors of ∠1 and ∠3 intersect at right angles.
14. Following a similar argument, we can prove that the bisectors of ∠2 and ∠4 intersect at right angles.
15. Hence, we have proved that the bisectors of the interior angles on the same sides of the transversal intersect at right angles.

Therefore, the bisectors of the interior angles on the same sides of a trans