Problem: Find the measure of angle CAD and angle DBC in a square ABCD with diagonals AC and BD intersecting at O.

Solution:

Step 1: Draw the square ABCD with diagonals AC and BD intersecting at O.

Step 2: Mark the angles CAD and DBC in the diagram.

Step 3: Use the properties of a square and diagonals to find the measure of angle CAD and angle DBC.

Step 4: Since ABCD is a square, all its sides are equal and all its angles are 90 degrees.

Step 5: Diagonals AC and BD bisect each other at O. Therefore, angle AOC and angle BOD are also 90 degrees.

Step 6: Since the diagonals of a square are equal, AC = BD.

Step 7: In triangle AOC, angle AOC is 90 degrees, angle OAC is 45 degrees (since AC is bisected by diagonal BD), and angle ACO is also 45 degrees (since all angles of a triangle add up to 180 degrees).

Step 8: Therefore, angle CAD is equal to angle OAC + angle ACO = 45 + 45 = 90 degrees.

Step 9: In triangle BOD, angle BOD is 90 degrees, angle OBD is 45 degrees (since BD is bisected by diagonal AC), and angle BDO is also 45 degrees (since all angles of a triangle add up to 180 degrees).

Step 10: Therefore, angle DBC is equal to angle OBD + angle BDO = 45 + 45 = 90 degrees.

Step 11: Hence, the measure of angle CAD and angle DBC in a square ABCD with diagonals AC and BD intersecting at O is 90 degrees.

Conclusion: Therefore, the measure of angle CAD and angle DBC in a square ABCD with diagonals AC and BD intersecting at O is 90 degrees.