Proof:

Given: We are given a parallelogram with two adjacent angles.

To prove: The angle bisectors of the two adjacent angles of a parallelogram bisect at 90 degrees.

Proof:

Let's consider a parallelogram ABCD, where AB || CD and AD || BC. We are given that angle A and angle B are adjacent angles in the parallelogram.

Step 1: Draw a parallelogram ABCD

Let's start by drawing a parallelogram ABCD.

A----------------B
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| |
| |
D----------------C

Step 2: Draw angle bisectors for angle A and angle B

Next, draw the angle bisectors for angle A and angle B. Let the angle bisector of angle A intersect BC at point E, and the angle bisector of angle B intersect AD at point F.

A----------------B
| /
| /
| /
D-----------E--C

Step 3: Prove that angle EAF is a right angle

To prove that angle EAF is a right angle, we need to show that angle EAF = 90 degrees.

Since AB || CD and AD || BC, we can conclude that triangle ABD and triangle BCD are similar by the AA similarity criterion.

Therefore, angle ABD = angle BCD.

Since angle EAB is the angle bisector of angle A, we have angle EAB = angle BAD.

Similarly, since angle EAD is the angle bisector of angle A, we have angle EAD = angle ACD.

Combining these equalities, we have angle BAD = angle ACD.

Therefore, angle BAD + angle ACD = 180 degrees (angles of a straight line).

Substituting the equalities, we have angle EAB + angle EAD = 180 degrees.

Since angle EAB = angle BAD and angle EAD = angle ACD, we can rewrite the equation as:

angle BAD + angle ACD = 180 degrees.

But angle BAD = angle EAF and angle ACD = angle FAE.

Therefore, angle EAF + angle FAE = 180 degrees.

Since the sum of angles in a triangle is 180 degrees, we can conclude that angle EAF = angle FAE = 90 degrees.

Thus, the angle bisectors of angle A and angle B in the parallelogram ABCD bisect at 90 degrees.