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Definition of Line

A line does not have any endpoints. It has an infinite length.
Introduction to Lines & Angles | Quantitative for GMAT

Definition of a Line Segment

A line segment is a segment of a line, or in other words, we can say that a line segment is a line with two endpoints.
For example, The diagram shows a line L and one segment of this line is AB.
In a plane, there can be many lines or line segments.
And, these lines can be divided into a few types based on the relative positioning of a line with another line.
Introduction to Lines & Angles | Quantitative for GMAT

Types of lines

Intersecting Lines

Two lines are intersecting lines if they meet each other at a common point.
For example, L1 and L2 are intersecting lines in the below diagram
Introduction to Lines & Angles | Quantitative for GMAT

Parallel Lines 

A pair of lines are parallel if they never intersect.
For example, L1, L2, and L3 are parallel lines in the below diagram.
Introduction to Lines & Angles | Quantitative for GMAT

Transversal Line 

A transversal line cuts two or more lines at distinct points.
For example, Line L3 is the transversal line in the below diagram.
Introduction to Lines & Angles | Quantitative for GMAT

Angle – What is it?

Introduction to Lines & Angles | Quantitative for GMAT

An angle is formed when two lines intersect each other. We represent an angle by the symbol ∠.
An angle involves two legs and one common vertex at which two lines meet.
For example, ∠AOD is formed when lines AB and CD intersect with each other.
Also, ∠AOD is formed between the leg AO and OD, so we include A, O, and D while naming the angle.

Measurement of an Angle

Introduction to Lines & Angles | Quantitative for GMAT

The angle is measured in degrees.
An angle can measure from zero (0) degrees to 360 degrees. Based on the measurement of an angle, they are divided into four types:

  • Acute angle
  • Right angle
  • Obtuse angle
  • Reflex angle

Acute Angle

When the measurement of the angle is between 0 degrees and 90 degrees.
Introduction to Lines & Angles | Quantitative for GMAT

Right Angle

Introduction to Lines & Angles | Quantitative for GMAT

When the measurement of the angle is exactly 90 degree.

  • If there is a right angle between two lines, then the two lines are said to be perpendicular to each other.

Obtuse Angle

When the measurement of the angle is between 90 degrees and 180 degrees.
Introduction to Lines & Angles | Quantitative for GMAT

A straight line has an angle of 180 degrees.

Reflex Angle
Introduction to Lines & Angles | Quantitative for GMAT

  • When the measurement of the angle is between 180 degrees and 360 degrees.
  • We have discussed the basic type of angles.
  • Let us now discuss the angles formed when two lines intersect each other.

Angles formed between two intersecting lines

Vertically Opposite Angles
When two lines intersect each other, then 4 angles are formed.

  • And, the angles that are opposite to each other at the intersection point are known as vertically opposite angles.
  • Vertically opposite angles are always equal.

Introduction to Lines & Angles | Quantitative for GMAT

Let us now discuss the angles formed when two lines are intersected by a third line i.e. a transversal line. 

Angles formed by a transversal line

Introduction to Lines & Angles | Quantitative for GMAT

When a transversal line intersects two lines, then eight angles are formed as shown.
Introduction to Lines & Angles | Quantitative for GMAT

Now, there are several special pairs of angles that are obtained from this diagram.
For example: If you notice  (∠1, ∠3), (∠2, ∠4), (∠5, ∠7), and (∠6, ∠8) are all vertically opposite angles.
Similarly, we get several other types of angles. Let us discuss them.

Other types of Angles

Interior and Exterior Angles

Interior angles are the ones that are present inside the region between two lines.

  • And, exterior angles are the ones that are not present inside this region.

For example:

  • ∠2, ∠3, ∠5, and ∠8 are interior angles.
  • And, ∠1, ∠4, ∠6, and ∠7 are exterior angles.

Corresponding Angles

Introduction to Lines & Angles | Quantitative for GMAT

Two angles are said to be corresponding angles if they lie on the same side of the transversal line such that:

  • One angle is an interior angle, and
  • Another is an exterior angle

For example:
(∠4, ∠8), (∠3, ∠7), (∠1, ∠5), and (∠2, ∠6) are 4 pairs of corresponding angles

Alternate interior angles

Introduction to Lines & Angles | Quantitative for GMAT

Two interior angles, present on the opposite side of a transversal line, are called alternate interior angles.
For example:

(∠2, ∠8) and (∠3, ∠5) are alternate interior angles.

Alternate exterior angles

Introduction to Lines & Angles | Quantitative for GMAT

Two exterior angles that are present on the opposite side of the transversal line are called alternate exterior angles.
For example:

  • (∠1, ∠7) and (∠4, ∠6) are alternate exterior angles.

Note: If a transversal line intersects two parallel lines, then the corresponding angles, alternate interior angles, and alternate exterior angles are equal. So, we have discussed all the type of angles. Let us now learn about a few properties of angles.

Properties of Angles

Sum of angles on one side of a straight line
Introduction to Lines & Angles | Quantitative for GMAT

The sum of all the angles on one side of a straight line is always 180 degrees.
For example, The sum of ∠1, ∠2, and ∠3 is 180 degrees.

Sum of angles around a point

Introduction to Lines & Angles | Quantitative for GMAT

The sum of all the angles around a point is always 360 degrees.

For example, Sum of angles (∠1, ∠2, and ∠3) around point O is 360 degrees.

The document Introduction to Lines & Angles | Quantitative for GMAT is a part of the GMAT Course Quantitative for GMAT.
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FAQs on Introduction to Lines & Angles - Quantitative for GMAT

1. What is a line in geometry?
Ans. A line in geometry is a straight path that extends infinitely in both directions. It is a one-dimensional object that has no thickness or width.
2. What are the different types of lines?
Ans. There are several types of lines in geometry, including: - Horizontal line: A line that is parallel to the horizon and does not slant. - Vertical line: A line that goes straight up and down, perpendicular to the horizon. - Diagonal line: A line that slants or leans at an angle. - Perpendicular line: Two lines that intersect at a right angle (90 degrees). - Parallel lines: Two lines that never intersect and are always the same distance apart from each other.
3. How can you determine if lines are parallel or perpendicular?
Ans. Two lines are parallel if they never intersect and are always the same distance apart. To determine if two lines are parallel, you can compare their slopes. If the slopes are equal, the lines are parallel. Two lines are perpendicular if they intersect at a right angle (90 degrees). To determine if two lines are perpendicular, you can calculate their slopes. If the slopes are negative reciprocals of each other (multiplying one slope by -1 and then flipping its sign), the lines are perpendicular.
4. Can lines have endpoints?
Ans. No, lines do not have endpoints. They extend infinitely in both directions, so they have no specific starting or ending points. However, line segments, which are portions of lines with two distinct endpoints, do have endpoints.
5. What is the importance of lines in geometry?
Ans. Lines are fundamental elements in geometry and are used to describe and analyze various shapes and figures. They serve as the basis for understanding angles, triangles, polygons, and other geometric concepts. Lines are also crucial in real-world applications, such as architecture, engineering, and design, where precise measurements and alignments are essential.
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