Kite ABCD with Diagonals AC and BD intersecting at O

Given that ABCD is a kite with diagonals AC and BD intersecting at point O. We are also given that angle ABO is 32 degrees and angle OCD is 40 degrees. Our task is to find angle BCD.

Properties of a Kite
- A kite is a quadrilateral with two pairs of congruent adjacent sides.
- The diagonals of a kite are perpendicular to each other.
- The diagonals of a kite bisect each other.

Using the Properties of a Kite
Since ABCD is a kite, we can use the properties of a kite to find the measure of angle BCD.

The Diagonals of a Kite are Perpendicular to Each Other
Since AC and BD are the diagonals of the kite, we know that they are perpendicular to each other. Therefore, angle AOB and angle COD are both right angles, each measuring 90 degrees.

The Diagonals of a Kite Bisect Each Other
The diagonals of a kite, AC and BD, bisect each other. This means that the line segment AO is congruent to line segment CO, and line segment BO is congruent to line segment DO.

Using Angle Relationships
Since AO and CO are congruent, and angle AOB is 32 degrees, we can conclude that angle COB is also 32 degrees. This is because angles opposite congruent sides are congruent.

Similarly, since BO and DO are congruent, and angle COD is 40 degrees, we can conclude that angle BOD is also 40 degrees.

Finding Angle BCD
To find angle BCD, we need to subtract the sum of angles BOC and BOD from 180 degrees, since angles BOC and BOD are adjacent angles on a straight line.

Angle BOC = angle AOB + angle COB = 32 degrees + 32 degrees = 64 degrees
Angle BOD = angle COD + angle BOD = 40 degrees + 40 degrees = 80 degrees

Therefore, angle BCD = 180 degrees - (angle BOC + angle BOD) = 180 degrees - (64 degrees + 80 degrees) = 36 degrees.

Hence, angle BCD measures 36 degrees.