Question

Angle Between Two Lines Find the angle of intersection between line \(L_{1}\) having an angle of inclination of \(22^{\circ}\) and line \(L_{2}\) having an angle of inclination of \(86^{\circ} .\)

Short answer

The angle of intersection between the two lines is \(64^{\text{\(\circ\)}}\).

Step-by-step solution

Understanding Angle of Inclination

The angle of inclination of a straight line is the angle formed with the positive direction of the x-axis. For lines in a plane, the angle of intersection is given by the difference in their angles of inclination, assuming no line is exactly vertical.

Calculating the Angle of Intersection

To calculate the angle of intersection, subtract the angle of inclination of the first line (\(L_{1}\)) from that of the second line (\(L_{2}\)). The result will be the difference in degrees between the two lines.

Performing the Calculation

Subtract the angle of inclination of line 1 from the angle of inclination of line 2: \(86^{\text{\(\circ\)}} - 22^{\text{\(\circ\)}} = 64^{\text{\(\circ\)}}\). Therefore, the angle of intersection between the two lines is \(64^{\text{\(\circ\)}}\).

Key concepts

Angle of Inclination

The angle of inclination is a fundamental concept in understanding how lines are oriented within a coordinate system. It is defined as the angle formed between a straight line and the positive direction of the x-axis. This angle is measured in a counterclockwise direction from the axis to the line.

Imagine you're standing on a flat field holding a flashlight pointing directly in front of you. That beam of light would be analogous to the x-axis. If you start tilting the flashlight upward, the angle it makes with the ground increases—this represents the angle of inclination of a line. Typical values range from 0 to 180 degrees. Lines with the same angle of inclination are parallel to each other, whereas lines with different angles will intersect at some point unless they are parallel and offset.

Calculating Angles

Calculating angles is a staple activity in geometry and trigonometry, and there are various methods to determine the values depending on the context. For instance, when dealing with the angle of intersection between two lines, the procedure involves finding the difference between their angles of inclination.

Here’s how to simplify the process:

• Ensure that all angles are measured with the same unit (degrees are commonly used).
• Determine the angle of inclination for each line, which is the angle between the line and the positive x-axis.
• Calculate the absolute difference between these angles to get the angle of intersection.

This approach holds as long as neither line is perpendicular to the x-axis, which would make its angle of inclination 90 degrees and require a different method of calculation.

Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. In geometry, understanding the position and intersection of lines is crucial for solving problems related to shapes, angles, and distances.

In the context of the angle of intersection, geometry provides us with the principles to understand how lines relate to one another within a plane. When two lines intersect, they form angles at their point of crossing. These angles are geometric facts that can be used to determine other elements of a structure, like the size of a triangle's interior angles or a polygon's exterior angles.

Trigonometry

Trigonometry is a segment of mathematics that explores the relationships between the angles and lengths of triangles. It's deeply interconnected with geometry, as angles are present everywhere in geometric figures.

In trigonometry, the concepts of sine, cosine, and tangent are essential for dealing with angles. While this exercise on calculating the angle of intersection between two lines doesn't directly use trigonometric ratios, the principles of measuring angles, understanding the unit circle, and the role of angles in trigonometry are important foundations. Trigonometric functions can be used for finding angles in more complex scenarios, such as when lines are not strictly horizontal or vertical, or when dealing with three-dimensional problems.