Solving for Angles of a Quadrilateral in Arithmetic Progression

When the angles of a quadrilateral are arranged in an arithmetic progression (AP), determining the individual angles given the smallest angle can be quite interesting. In this article, we will go through the step-by-step process to solve for these angles, ensuring a clear understanding and a practical approach for readers. The sum of the angles in any quadrilateral is always 360°. We will begin by setting up the problem and solving it step-by-step.

Problem: The smallest angle is 15°, and angles are in AP.

Given the smallest angle as a 15°, we need to find the angles when they are in AP.

We can denote the angles as:

First angle: a 15° Second angle: a d Third angle: a 2d Fourth angle: a 3d

The sum of these angles is 360°. Therefore, we can set up the equation as:

(text{a} (text{a} text{d}) (text{a} text{2d}) (text{a} text{3d}) 360°)

(text{4a}   6text{d}  360°)

Substitute a 15° into the equation:

(4 times 15°   6text{d}  360°)

Following the steps:

(60°   6text{d}  360°)

Solving for d:

(6text{d}  300°)

(text{d} 50°)

Now, we can find the angles:

First angle: a 15° Second angle: a d 15° 50° 65° Third angle: a 2d 15° 2 × 50° 115° Fourth angle: a 3d 15° 3 × 50° 165°

The angles of the quadrilateral are therefore 15°, 65°, 115°, and 165°.

Another Approach:

To confirm, another method can be used:

Using AP properties: a, a d, a 2d, a 3d Sum of angles 360° Setting up the equation: 15° (15° d) (15° 2d) (15° 3d) 360°

Following the steps:

(4 times 15°   6text{d}  360°)

Leading to: 6d 300°, thus d 50°.

Therefore, the angles are 15°, 65°, 115°, and 165°.

Conclusion:

The angles of the quadrilateral are 15°, 65°, 115°, and 165°. This solution is consistent with the properties of an arithmetic progression and the sum of the angles in a quadrilateral.

Key Points:

The angles are in AP with the smallest angle being 15°. The common difference is found to be 50°. The angles are then calculated using the arithmetic progression properties.

Further Reading:

For a deeper understanding of the properties of quadrilaterals and arithmetic progressions, you may find the following articles useful:

Understanding Quadrilaterals Solving Problems with Arithmetic Progressions