Lines and Angles - Properties, Types, Examples

Lines and Angles

Lines and Angles are the basic terms used in geometry and form the foundation for understanding all geometric concepts. A line is a one-dimensional figure that extends infinitely in both directions. It has length but no width or thickness.

An angle is the opening formed when two or more lines or rays meet at a common point called the vertex. Angles are measured in degrees or radians. Every geometric figure is made up of lines and angles, and learning about them helps us understand shapes, measurements, and patterns in geometry. 

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What are Lines?

A line is a straight, one-dimensional figure that extends infinitely in both directions and is made up of countless points placed next to each other. A line has length but no width or thickness, which is why it is described as one dimensional.  The length of a line cannot be measured because it has no endpoints.

If a figure has both a starting and an ending point, it is known as a line segment, and its length can be measured. When a line begins at a fixed point and extends infinitely in one direction, it is called a ray. 

Read More: Basic Geometrical Ideas

Properties of Lines

Below are five key properties of lines that explain their nature and behaviour:

1. A line extends infinitely in both directions.  It has no starting or ending point, continuing endlessly on both sides.

2. A line has only one dimension, which is length.  It does not have width or thickness.

3. Three or more points lying on the same line are called collinear points.  If points A, B, and C are on the same line, they are said to be collinear.

4. Two lines that never meet and remain the same distance apart are called parallel lines.
   For example, the edges of a ruler or railway tracks are parallel lines.

5. Two lines that meet or intersect at a right angle (90°) are called perpendicular lines.
   The corners of a square or the meeting of a wall and floor show perpendicular lines.

Types of Lines

Here are the different types of lines you will find in geometry:

Horizontal Line: A line that runs from left to right, like the horizon. It represents flatness and stability, seen in tables or window edges.

Vertical Line: A straight line running from top to bottom, such as walls or poles. It shows height and strength.

Slanting Line: A line tilted at an angle, neither vertical nor horizontal, found in roofs, ladders, or ramps.

Parallel Lines: Lines that move in the same direction and never meet, like railway tracks or road markings.

Perpendicular Lines: Lines meeting at a right angle (90°), forming neat corners in doors, tiles, and books.

Intersecting Lines: Lines that cross each other at a point, forming angles, like the letter X or the intersection of roads.

What are Angles?

An angle is a figure formed when two straight lines, rays, or line segments meet at a common point. This common meeting point is called the vertex of the angle. The two lines that form the angle are known as the arms or sides of the angle.

The amount of turning or opening between the two arms determines how wide or narrow the angle is. This turning is what we call the measure of the angle. Angles are measured in degrees (°), and a complete turn makes an angle of 360°.

Types of Angles

Below are the main kinds of angles you will come across in geometry.

1. Acute Angle

An acute angle measures less than 90°. It is sharp and narrow, like the pointed tip of a triangle or the open lid of a pencil box that is lifted only slightly. 

2. Right Angle

A right angle measures exactly 90°. It looks like the corner of a book, a square, or the meeting point of a wall and the floor. 

3. Obtuse Angle 

An obtuse angle is greater than 90° but less than 180°. It is wider than a right angle but smaller than a straight line. 

4. Straight Angle

A straight angle measures exactly 180°. It looks like a straight line because both arms extend in opposite directions from the vertex. 

5. Reflex Angle

A reflex angle measures more than 180° but less than 360°. It looks like the larger opening that remains when you measure the outside of an acute or obtuse angle. For example, if the hands of a clock show 5 o’clock, the larger gap between them forms a reflex angle.

6. Complete Angle

A complete angle measures exactly 360°. It represents one full rotation or complete turn of a ray around its vertex. For example, when the hands of a clock complete one full circle, they make a complete angle.

Lines and Angles in Geometry

Geometry is built upon lines and angles. Every triangle, square, rectangle, and polygon is made up of lines meeting at certain angles. When studying lines and angles properties, it helps to know how they interact and relate to each other. Some key lines and angles properties include:

1. When two lines intersect, opposite (or vertical) angles formed are equal.

2. The sum of angles on a straight line is always 180°.

3. The sum of angles around a point is always 360°.

4. If two lines are parallel and a third line cuts them, corresponding angles are equal.

5. Alternate interior angles formed by a transversal are also equal when lines are parallel.

These lines and angles properties make it easier to solve geometry questions and understand shapes more deeply.

Lines and Angles Formulas

Knowing a few basic lines and angles formulas makes geometry much easier to understand. These formulas help you find unknown angles and check if lines are parallel.

1.  All angles formed on a straight line always add up to 180°.

2.  All angles that meet at one point add up to 360°.

3.  When two lines cross, the angles opposite each other are always equal.

4. If Two Lines are Parallel:

• Corresponding Angles are Equal

• Alternate Interior Angles are Equal

• Co-interior Angles are Supplementary (Sum = 180°)

Read More: Points and Lines

Lines and Angles Examples

Example 1:  Two lines intersect each other. One of the angles formed is 70°. What is the measure of its vertically opposite angle?

Solution: When two lines cross, vertically opposite angles are always equal. Therefore, the other angle will also measure 70°.

Example 2: If two parallel lines are cut by a transversal and one of the corresponding angles is 60°, what is the measure of the other corresponding angle?

Solution: Corresponding angles formed by a transversal cutting parallel lines are equal. Hence, the other corresponding angle is also 60°.

Example 3: The sum of angles on a straight line must be 180°. If two angles are 50° and 80°, find the third angle.

Solution: Sum of three angles = 180° So, the third angle = 180° − (50° + 80°) = 50°.

Example 4: Two parallel lines are intersected by a transversal. One of the alternate interior angles is 75°. Find the other alternate interior angle.

Solution: Alternate interior angles between parallel lines are equal. Therefore, the other alternate interior angle is also 75°.

Lines and Angles Fun Facts

Here are a few fun facts that make learning about lines and angles even more interesting:

1. The letter L shows a perfect right angle.

2. The letter X has intersecting lines that form equal opposite angles.

3. A triangle is the simplest shape made of three line segments and three angles.

4. The sum of the three angles inside any triangle is always 180°.

These simple facts connect lines and angles properties with shapes we see in daily life, helping us understand how geometry appears in real objects all around us.

Also read:  How to Find the Angle of a Triangle?

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