Introduction:

To prove that the bisector of any two consecutive angles of a parallelogram intersect at a right angle, we need to establish certain properties of parallelograms and angles. It is important to understand the definition of a parallelogram and the properties of its angles before proceeding with the proof.

Properties of a Parallelogram:

1. Opposite sides of a parallelogram are equal in length.
2. Opposite angles of a parallelogram are equal in measure.
3. Consecutive angles of a parallelogram are supplementary, meaning their measures add up to 180 degrees.

Proof:

Let's consider a parallelogram ABCD, where AB is parallel to CD and AD is parallel to BC.

Step 1: Draw the diagonals AC and BD.

Step 2: Let's assume the angles A and B are the consecutive angles of the parallelogram.

Step 3: Draw the bisectors of angles A and B, intersecting at point O.

Step 4: To prove that the bisectors intersect at a right angle, we need to show that angle AOC is 90 degrees.

Step 5: Since ABCD is a parallelogram, opposite angles are equal. Therefore, angle A is equal to angle C, and angle B is equal to angle D.

Step 6: The bisectors of angles A and B divide them into two equal parts.

Step 7: Therefore, angle AOC is equal to angle COB, and angle COD is equal to angle BOD.

Step 8: The sum of angle AOC and angle COB is equal to angle AOB, which is 180 degrees (consecutive angles of a parallelogram are supplementary).

Step 9: Since angle AOC is equal to angle COB, their sum is 180 degrees divided by 2, which is 90 degrees.

Step 10: Hence, the bisectors of angles A and B intersect at a right angle.

Conclusion:

The bisector of any two consecutive angles of a parallelogram intersect at a right angle. This is established by using the properties of parallelograms, which state that opposite angles are equal and consecutive angles are supplementary. By dividing the consecutive angles into two equal parts, the bisectors form angles that add up to 180 degrees when combined. Since these angles are equal, their sum is 90 degrees, proving that the bisectors intersect at a right angle.