Proof that the bisectors of the angles of a linear pair are at right angles:

To prove that the bisectors of the angles of a linear pair are at right angles, we need to understand the properties of a linear pair and angle bisectors.

Linear Pair:
A linear pair is formed when two adjacent angles are supplementary, meaning their sum is equal to 180 degrees. In a linear pair, the angles share a common vertex and a common side, while their other sides form a straight line.

Angle Bisectors:
An angle bisector is a line or ray that divides an angle into two equal parts. It cuts the angle into two congruent angles. The angle bisector intersects the angle at its vertex.

Proof:

Step 1: Let there be a linear pair with two adjacent angles, ∠PQR and ∠RQS, sharing a common side QR.

Step 2: Assume that the bisector of ∠PQR is line AB, and the bisector of ∠RQS is line CD.

Step 3: As per the definition of an angle bisector, line AB divides ∠PQR into two equal angles, ∠PQA and ∠AQR. Similarly, line CD divides ∠RQS into two equal angles, ∠RQC and ∠CQS.

Step 4: Since line AB is the bisector of ∠PQR, it divides the angle into two congruent angles, ∠PQA and ∠AQR.

Step 5: Similarly, line CD is the bisector of ∠RQS and divides the angle into two congruent angles, ∠RQC and ∠CQS.

Step 6: Now, we can see that ∠PQA ≅ ∠CQS and ∠AQR ≅ ∠RQC. This is because the bisectors divide the angles into congruent parts.

Step 7: According to the definition of congruent angles, if two angles are congruent, their measures are equal.

Step 8: Therefore, we can conclude that ∠PQA = ∠CQS and ∠AQR = ∠RQC.

Step 9: By combining these two equalities, we get ∠PQA + ∠AQR = ∠CQS + ∠RQC.

Step 10: Since ∠PQR and ∠RQS are supplementary angles in a linear pair, their sum is equal to 180 degrees. Therefore, ∠PQR + ∠RQS = 180 degrees.

Step 11: By substituting the values from Step 9 and Step 10, we have 180 degrees = ∠CQS + ∠RQC.

Step 12: We can rearrange the equation to get ∠CQS + ∠RQC = 180 degrees.

Step 13: This equation shows that the sum of ∠CQS and ∠RQC is equal to 180 degrees, which proves that they are supplementary angles.

Step 14: By the definition of supplementary angles,