Question

Angle Between Two Lines Find the angle of intersection between line \(L_{1}\) having an angle of inclination of \(35^{\circ}\) and line \(L_{2}\) having an angle of inclination of \(160^{\circ}\).

Short answer

The angle of intersection between the two lines is |35 - 160| = 125 degrees.

Step-by-step solution

Understanding the Concept of Angle of Intersection

The angle of intersection between two lines is the smallest of the angles formed between the two lines. If we have the angles of inclination for both lines, we can find the angle of intersection by calculating the absolute value of the difference between these angles.

Calculating the Difference between the Angles of Inclination

Calculate the difference in the angles of inclination for line 1 and line 2. Since line 1 has an inclination of 35 degrees and line 2 has an inclination of 160 degrees, subtract one from the other to find the difference, ensuring to take the absolute value. The difference is |35 degrees - 160 degrees|.

Determining the Angle of Intersection

Find the absolute difference between the angles calculated in the previous step. Subtract to find the angle of intersection, which is |35 - 160| degrees.

Key concepts

Angle of Inclination

The angle of inclination is a foundational concept in geometry relevant to understanding how lines are oriented in a plane. It is defined as the acute angle a line makes with the positive direction of the x-axis. This angle can range from 0 to 90 degrees. For example, a horizontal line has an angle of inclination of 0 degrees, while a vertical line has an angle of inclination of 90 degrees.

The angle of inclination can be used to determine the direction and steepness of a line. This is particularly useful when comparing two or more lines, as in the case of the angle of intersection between lines. It helps to visualize the lines rising from the x-axis at their respective angles. The greater the angle of inclination, the steeper the line rises from the x-axis.

Calculating Angles in Geometry

Geometry often involves calculating angles, whether they be angles of inclination, angles of intersection, or various angles formed within shapes. To calculate the angle of intersection between two lines, one typically finds the difference between their angles of inclination. This is an application of understanding how to deal with angles numerically.

For instance, when given two lines with known angles of inclination, we find the angle of intersection by subtracting one angle from the other and taking the absolute value. It's crucial to consider the acute angle formed between the two lines, which is why we focus on the smallest positive angle when they intersect. This reinforces the concept of absolute value when dealing with the difference between angles.

Understanding Geometric Concepts

A profound understanding of geometric concepts is essential for solving problems involving angles, lines, and shapes. Grasping the respective definitions, properties, and relationships among geometric elements allows for effective problem-solving and logical reasoning.

When approaching a problem, it is important to visualize and analyze the given information. For angles of intersection, realizing that it refers to the smallest angle at which two lines meet enhances comprehension. Furthermore, employing known formulas and theorems can guide the process, as with the use of the angle of inclination to quantify a line's slant with respect to the x-axis. This interplay among different concepts highlights the importance of a broad understanding of geometry when encountering various problems.