Determining the Unknown Angle in a Quadrilateral

Understanding the angles and their properties within a quadrilateral is a fundamental concept in geometry. A quadrilateral is a polygon with four sides, and the sum of its interior angles is always 360 degrees. This property is essential for solving various geometric problems, such as determining the measure of an unknown angle given the other angles.

The Sum of Interior Angles in a Quadrilateral

A key fact in geometry is that the sum of the interior angles of any quadrilateral is 360 degrees. This is a constant value that applies to all quadrilaterals, whether they are convex or concave. Given the angles 90°, 80°, and 125° in a quadrilateral, we can determine the fourth angle, denoted as x, by using this fundamental property.

Solving for the Unknown Angle

Let's walk through the steps to find the value of x:

Step 1: Sum of Known Angles

The sum of the known angles is:

[ 90^circ 80^circ 125^circ 295^circ ]

Step 2: Subtract from the Total Angle Sum

To find the value of the unknown angle x:

[ x 360^circ - 295^circ ]

Step 3: Calculate the Value of x

Perform the subtraction to get the value of x:

[ x 65^circ ]

Thus, the value of the fourth angle is 65 degrees.

Exterior Angles of a Quadrilateral

It's also important to understand that the sum of the exterior angles of any convex polygon, including a quadrilateral, is always 360 degrees. This property is useful for verifying the correct sum of exterior angles and can provide additional insight into the geometric properties of the figure.

Interior and Exterior Angles Relationship

Another interesting fact is that the interior angle plus its corresponding exterior angle always equals 180 degrees. This relationship can be used as a supplement to the sum of the interior angles property. For example, in the given quadrilateral:

The exterior angle corresponding to the 90° interior angle is 90°. The exterior angle corresponding to the 80° interior angle is 100°. The exterior angle corresponding to the 125° interior angle is 55°. The exterior angle corresponding to the 65° interior angle is 115°.

Together, they sum to 360 degrees:

[ 90^circ 100^circ 55^circ 115^circ 360^circ ]

Conclusion

In summary, the measure of the unknown angle x in a quadrilateral with angles 90°, 80°, and 125° is 65°. This can be determined by subtracting the sum of the known angles from the total sum of the interior angles of a quadrilateral, which is 360 degrees. Understanding these properties can help solve a wide range of geometric problems and is a valuable tool in both academic and real-world applications.