## Key Concepts

- Name points, lines, and planes
- Name segments, rays, and opposite rays
- Sketch intersections of lines and planes

## Essentials of Geometry

### Introduction

In this chapter, we will learn about name points, lines, planes, name segments, rays, opposite rays, sketching intersections of lines and planes.

### Identify Points, Lines, and Planes

Points, lines and planes are the basic concepts of geometry and can be found in many real-life examples.

A location of a place on the map is a point.

The center-line on a highway and the equator on the map are lines.

A piece of paper and a whiteboard are examples of a plane.

### Name points, lines, and planes

Name points, lines, and planes do not have any formal definitions. These are undefined terms. They are basic geometric structures. Any shape created in geometry is based on these three terms.

#### What is a point?

A point has no dimension, and it is represented by a dot. It helps us to show the location. Points do not have any actual size. For naming points, we use capital letters like A, B, C, etc.

**Example:**

#### What is a line?

Lines are straight paths that extend in two opposite directions without end. They are made up of an infinite amount of points.

A line can be represented in two ways:

- By using the 2 points on the line.
- By a lower-case letter.

**Example:**

The above line can be represented with the letter ‘l.’

Or

It can be represented by using the 2 name points like,

Or

#### What is a plane?

A plane is a flat two-dimensional surface that extends without end in all four directions. Planes are made up of an infinite amount of points.

A plane can be represented in two ways:

- By using the 3 points on the lines.
- By a capital letter.

**Example:**

The above plane can be represented with the letter ‘G.’

Or

It can be represented by using the 3 name points like,

Plane DEF.

#### What are collinear points?

The points that are on the same line are called collinear points.

**Example:**

In the above example, A, B, and C are collinear points because they are on the same line.

#### What are coplanar points?

The points that are on the same plane are called coplanar points.

In the above example, A, B, and C are coplanar points because they are on the same plane.

**Example 1:**

Let us understand more about name points, lines, and planes.

Look at the given plane ‘R.’

**1. Give two other names for** .

We can name it as

Or

Line b

**2. Give two other names for plane R**.

H J I

Or

G J I

**3. Name three collinear points. **

H, G, I because these points are on the same line.

**4. Name four coplanar points. **

H, G, J, I because these points are on the same plane.

### Name segments, rays, and opposite rays

#### What is a line segment?

The line segment has two endpoints and cannot extend further.

**Example:**

The above line segment can be represented as:

Or

#### What is a ray?

A ray has one endpoint, which is called the initial point, and it can extend out in one direction without an end.

**Example:**

The above ray can be represented as:

because the initial point is C and is extending through point D.

#### What are opposite rays?

Opposite rays are the two rays, which has the same initial point but extends in opposite directions. They look like a line.

**Example:**

The above opposite rays can be represented as:

Because E is the initial point and F, G are endpoints.

**Example 1:**

Let us understand more about segments, rays, and opposite rays.

Look at the given picture.

**1. Give another name for** .

We can name it as .

**2. Name all the rays with endpoint K. **

The rays that have K as an endpoint are,

**3. Which pairs are opposite rays? **

The opposite rays are,

### Sketch intersections of lines and planes

#### What is an intersection?

If two lines intersect at one point, it is called an intersection. The intersection of the figures is the set of points the figures have in common.

**Example 1:**

In the above example, the red line and blue line intersect each other at one point.

**Example 2:**

Let us sketch a plane and a line in that plane:

**Example 3:**

Let us sketch a plane and a line that intersects the plane at one point:

**Example 4:**

Sketch a plane and a line that does not intersect the plane**. **

**Example 5:**

In this example, *x* is the point of intersection of and .

**Example 6:**

In this example, a line and a plane are intersecting at one point.

**Example 7:**

In this example, two planes intersect each other at a line.

**Example 8:**

Let us sketch two planes that intersect in a line.

## Exercise

- Draw and label each of the following

- AB
^{–} - Points C and D
- RS
^{–}

- Draw and label each of the following

- Points L, M, and N
- MN
^{–} - JK
^{–}

- Use the figure below and answer the following questions
- Name four points.
- Name two line segments.
- Name three rays.
- Name the line three ways.

- Use the figure below and answer the following questions
- Name the point of intersection.
- Name the two lines that intersect.

- Draw and label each of the following

a. LM intersects NO at point P.

b. Y is the point at which XZ intersects WV - Identify whether the following points are collinear or coplanar

- Identify whether the following points are collinear or coplanar.

- Use the figure below and answer the following questions.
- Name all sets of collinear points.
- Name the points that are not collinear to . DE

- Use the plane below and answer the following questions

- Examples of points: _______________________
- Example of line: __________________________
- Examples of segments: _____________________
- Examples of rays: ________________________

- What kind of geometric intersection do the photographs suggest?

### What have we learned

- About name points, lines, planes
- Collinear points and coplanar points
- Name segments, rays, opposite rays
- Sketching intersections of lines and planes

### Concept Map

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