Understanding the Problem
To find the acute angle between the diagonals of a rectangle when one diagonal is inclined at 50 degrees to one side, we can use some basic properties of rectangles and angles.
Properties of a Rectangle
- A rectangle has two pairs of equal sides.
- The diagonals of a rectangle bisect each other and are equal in length.
- The angles formed by the diagonals with the sides are congruent.
Diagram Analysis
- Let’s label the rectangle as ABCD, with sides AB and BC.
- The diagonal AC is inclined at 50 degrees to side AB.
Finding the Angle Between the Diagonals
- The other diagonal, BD, also meets the sides at certain angles.
- Since diagonals bisect each other, if diagonal AC makes an angle of 50 degrees with AB, then diagonal BD makes an angle of 90 degrees - 50 degrees with side AB, which equals 40 degrees.
Calculating the Angle Between Diagonals
- The acute angle between the diagonals can be found by taking the sum of the angles made by each diagonal with the sides of the rectangle.
- The angle between diagonals AC and BD is given by:
Angle between diagonals = 50 degrees + 40 degrees = 90 degrees
- Since we need the acute angle, we take the smaller angle, which is 90 degrees.
Conclusion
- The acute angle between the diagonals of the rectangle is 90 degrees.
This means that the diagonals of the rectangle intersect at right angles, which is a unique property of rectangles.