Geometric Figures and Spatial Reasoning

Points, Lines, Segments, and Rays are the basic building blocks of geometry. When you understand these, it becomes much easier to describe shapes, draw figures, and talk about how parts of a shape connect.

In this lesson, you will learn what a point, line, line segment, and ray are. You will also learn how to tell them apart and how to name them correctly.

1. What is a point?

A point shows an exact place or location. A point does not have any length or width. It is usually shown as a small dot.

We name points with capital letters, like Point A, Point B, or Point C.

For example, if you see a dot labeled A, you would call it Point A.

2. What is a line?

A line is a straight path that goes on forever in both directions. Because it never ends, a line has no endpoints.

When we draw a line on paper, we usually show arrows on both ends to remind us that the line keeps going.

A line can be named using any two points on it. For example, a line through points \(A\) and \(B\) can be called line \(AB\).

We can write it like this:

\(\overleftrightarrow{AB}\)

The arrows over the letters show that the line goes forever in both directions.

3. What is a line segment?

A line segment is part of a line. It has two endpoints. Because it has endpoints, a line segment has a fixed length.

If a segment starts at point \(A\) and ends at point \(B\), it is called segment \(AB\).

We can write it like this:

\(\overline{AB}\)

The bar over the letters shows that it is a segment. Unlike a line, a segment does not go on forever.

4. What is a ray?

A ray is a straight path that starts at one point and goes on forever in one direction.

A ray has one endpoint. The other side keeps going forever.

If a ray starts at point \(A\) and goes through point \(B\), it is called ray \(AB\).

We can write it like this:

\(\overrightarrow{AB}\)

This means the ray begins at \(A\) and passes through \(B\). The order of the letters matters for rays because the first letter names the endpoint.

How are they different?

The easiest way to tell these apart is to look at how many endpoints each one has.

• Point: an exact location
• Line: 0 endpoints, goes forever both ways
• Line segment: 2 endpoints
• Ray: 1 endpoint, goes forever one way

Here is another way to think about them:

• A point is like a dot on a map.
• A line is like a road that never ends in either direction.
• A line segment is like the part of a road between two stops.
• A ray is like a flashlight beam that starts at the flashlight and keeps shining forward.

How to name them

It is important to name each figure correctly.

1. Point: use one capital letter, such as \(A\).
2. Line: use two points on the line, such as \(\overleftrightarrow{CD}\).
3. Line segment: use its two endpoints, such as \(\overline{EF}\).
4. Ray: use the endpoint first, then another point on the ray, such as \(\overrightarrow{GH}\).

Remember: for a ray, the order matters. \(\overrightarrow{GH}\) is not the same as \(\overrightarrow{HG}\).

Worked Example 1: Identify the figure

A drawing shows a straight path with arrows on both ends and points \(M\) and \(N\) on it. What is it?

Step 1: Look at the ends. There are arrows on both ends.

Step 2: A figure with arrows on both ends goes forever both ways.

Answer: It is a line, named \(\overleftrightarrow{MN}\).

Worked Example 2: Segment or ray?

A figure starts at point \(P\), goes through point \(Q\), and has an arrow only on the side past \(Q\). What is it?

Step 1: It has one endpoint, at \(P\).

Step 2: It keeps going in one direction.

Answer: It is a ray, named \(\overrightarrow{PQ}\).

Worked Example 3: Count endpoints

A figure connects point \(R\) to point \(S\) with no arrows. What is it?

Step 1: It has two endpoints: \(R\) and \(S\).

Step 2: A figure with two endpoints is a line segment.

Answer: It is segment \(RS\), written \(\overline{RS}\).

Worked Example 4: Name the ray correctly

A ray begins at \(A\) and goes through \(B\). Should it be named \(\overrightarrow{AB}\) or \(\overrightarrow{BA}\)?

Step 1: Find the endpoint. The endpoint is \(A\).

Step 2: The endpoint must be named first.

Answer: The correct name is \(\overrightarrow{AB}\).

Tips for drawing these figures

• Draw a point as a dot and label it with a capital letter.
• Draw a line as a straight path with arrows on both ends.
• Draw a line segment as a straight path with two endpoints.
• Draw a ray as a straight path with one endpoint and one arrow.

Common mistakes to avoid

• Do not call a segment a line. A line never ends, but a segment has two endpoints.
• Do not forget that a ray has only one endpoint.
• Do not mix up the order when naming a ray. The endpoint comes first.
• Do not name a point with a lowercase letter. Points are usually named with capital letters.

Quick check

• If a figure has 0 endpoints, what is it? A line.
• If a figure has 2 endpoints, what is it? A line segment.
• If a figure has 1 endpoint, what is it? A ray.
• If it is just a dot showing a location, what is it? A point.

Summary

A point is an exact location. A line goes on forever in both directions. A line segment has two endpoints, and a ray has one endpoint and goes on forever in one direction.

When you look at a figure, count the endpoints. That is the best clue for deciding whether it is a line, a segment, or a ray. Then name it carefully using the correct letters and symbols.

Lesson: Understanding Angles and Rotation

In geometry, an angle is made when two rays meet at the same endpoint.

A ray is a straight path that starts at one point and goes on in one direction. The shared endpoint where the two rays meet is called the vertex.

You can think of an angle like the opening of a door, the hands of a clock, or the corner of a book. The wider the opening, the larger the angle.

An angle is not about how long the sides are. It is about how much turn or rotation there is from one ray to the other.

What does rotation mean?

Rotation means a turn. If one ray stays still and the other ray turns around the vertex, it makes different angles.

A small turn makes a small angle. A bigger turn makes a bigger angle.

So, an angle can be understood as the amount of turning from one side to the other.

Parts of an Angle

• 2 rays — the sides of the angle
• 1 vertex — the point where the rays meet
• inside space — the opening between the rays

If we name an angle, we often use the vertex letter in the middle. For example, in angle \(\angle ABC\), the vertex is point \(B\).

Angles Show Turn

Imagine an arrow pointing to the right. If it turns upward, it has made a quarter-turn. That turn forms an angle.

Here are some common turns:

• Quarter-turn — one of four equal turns
• Half-turn — two quarter-turns
• Full-turn — all the way around back to where it started

These turns can be measured in degrees:

• Quarter-turn = \(90^\circ\)
• Half-turn = \(180^\circ\)
• Full-turn = \(360^\circ\)

The symbol for degrees is \(^\circ\).

Important Angle Sizes

There are some angle sizes you should know well.

• Right angle — exactly \(90^\circ\)
• Straight angle — exactly \(180^\circ\)
• Full angle — exactly \(360^\circ\)

A right angle looks like the corner of a square or rectangle. A straight angle looks like a straight line. A full angle is one complete turn.

Comparing Angles

We can compare angles by looking at how much they open.

• If one angle has a smaller opening, it has less rotation.
• If one angle has a wider opening, it has more rotation.

Remember: longer sides do not mean a bigger angle. Only the amount of turn matters.

Example: Two angles can have sides of different lengths, but if the opening is the same, the angles are the same size.

Worked Example 1: Find the vertex

Look at \(\angle PQR\).

The middle letter tells us the vertex. So the vertex is Q.

Answer: The vertex of \(\angle PQR\) is \(Q\).

Worked Example 2: Identify the turn

A pointer starts by facing up. Then it turns to face right.

This is a quarter-turn. A quarter-turn makes a right angle.

$$90^\circ$$

Answer: The turn is a quarter-turn, and the angle is \(90^\circ\).

Worked Example 3: Compare two angles

Angle A is a quarter-turn. Angle B is a half-turn.

We know:

• Quarter-turn = \(90^\circ\)
• Half-turn = \(180^\circ\)

Since \(180^\circ > 90^\circ\), Angle B is larger.

Answer: Angle B is larger because it has more rotation.

Worked Example 4: Think about a full turn

A spinner points left. It makes one complete turn and points left again.

One complete turn is called a full-turn.

$$360^\circ$$

Answer: The spinner turned \(360^\circ\).

How to Recognize Angles in Real Life

You can see angles and rotation in many everyday objects:

• a door opening and closing
• clock hands turning
• scissors opening
• the corner of a room
• a spinning wheel

When something turns, it creates a change in angle.

Tips to Remember

• An angle is made by two rays sharing one endpoint.
• The shared endpoint is the vertex.
• An angle measures turn or rotation.
• A bigger turn means a bigger angle.
• Side length does not change the angle size.
• \(90^\circ\) is a quarter-turn, \(180^\circ\) is a half-turn, and \(360^\circ\) is a full-turn.

Brief Summary

An angle is the space made when two rays meet at one endpoint. That endpoint is called the vertex.

Angles tell us how much turning or rotation has happened from one ray to the other. We can describe turns as quarter-turns, half-turns, and full-turns, and we can measure them as \(90^\circ\), \(180^\circ\), and \(360^\circ\).

When you look at an angle, focus on the opening, not the side lengths. The opening shows the amount of turn.

Classifying Acute, Right, and Obtuse Angles

Have you ever noticed the corners of a book, a clock, or an open door? Those corners are called angles. In math, angles help us describe how much two lines open apart from the same point.

In this lesson, you will learn how to tell whether an angle is acute, right, or obtuse. A very important idea is to compare angles to a right angle, which measures exactly \(90^\circ\).

What is an angle?

An angle is made when two rays meet at one endpoint. That endpoint is called the vertex.

You can think of an angle like the opening of scissors or the opening of a door. The wider or smaller the opening is, the different the angle is.

The special angle to remember: the right angle

A right angle is an angle that measures exactly \(90^\circ\).

It looks like the corner of a square or rectangle. Many things around you have right angles, such as:

• the corner of a book
• the corner of a window
• the corner of a floor tile

Because a right angle is exactly \(90^\circ\), we use it as our comparison angle.

Here is the key idea:

• If an angle is smaller than \(90^\circ\), it is acute.
• If an angle is exactly \(90^\circ\), it is right.
• If an angle is greater than \(90^\circ\), it is obtuse.

We can write this like this:

1. Acute angles

An acute angle is a small angle. It opens less than a right angle.

If you compare an acute angle to the corner of a square, the acute angle will be narrower.

Examples of acute angles might look like:

• the tip of a slice of pizza
• the hands of a clock at a small opening
• a slightly opened pair of scissors

Some acute angle measurements are \(20^\circ\), \(45^\circ\), and \(89^\circ\).

2. Right angles

A right angle measures exactly \(90^\circ\).

It is not smaller than \(90^\circ\), and it is not larger than \(90^\circ\). It is exactly \(90^\circ\).

Right angles are often marked with a small square in the corner to show that the angle is a right angle.

Some examples are:

• the corner of a sheet of paper
• the corner of a picture frame
• where one wall meets the floor

3. Obtuse angles

An obtuse angle is bigger than a right angle. It opens more than \(90^\circ\).

If you compare an obtuse angle to the corner of a square, the obtuse angle will be wider.

Some obtuse angle measurements are \(100^\circ\), \(120^\circ\), and \(150^\circ\).

An obtuse angle is still not a straight line. It is just wider than a right angle.

How to classify an angle

To classify an angle means to decide which group it belongs to.

Use these steps:

1. Look at the angle carefully.
2. Think about a right angle, which is \(90^\circ\).
3. Ask: Is this angle smaller than, equal to, or greater than \(90^\circ\)?
4. Name it as acute, right, or obtuse.

Helpful clues

• Acute = smaller opening
• Right = square corner
• Obtuse = wider opening

Worked Example 1

An angle measures \(35^\circ\). What kind of angle is it?

Step 1: Compare \(35^\circ\) to \(90^\circ\).

Step 2: Since \(35^\circ < 90^\circ\), the angle is smaller than a right angle.

Answer: It is an acute angle.

Worked Example 2

An angle measures \(90^\circ\). What kind of angle is it?

Step 1: Compare \(90^\circ\) to \(90^\circ\).

Step 2: Since the angle is exactly \(90^\circ\), it matches a right angle.

Answer: It is a right angle.

Worked Example 3

An angle measures \(125^\circ\). What kind of angle is it?

Step 1: Compare \(125^\circ\) to \(90^\circ\).

Step 2: Since \(125^\circ > 90^\circ\), the angle is larger than a right angle.

Answer: It is an obtuse angle.

Worked Example 4

You see an angle that looks like the corner of a book. What kind of angle is it?

A book corner is shaped like a square corner. A square corner is a right angle.

Answer: It is a right angle.

Comparing angles without a number

Sometimes you will not be told the exact number of degrees. That is okay. You can still classify the angle by looking at how open it is.

• If it looks smaller than a square corner, it is acute.
• If it looks exactly like a square corner, it is right.
• If it looks wider than a square corner, it is obtuse.

Common mistakes to avoid

• Do not guess by the length of the sides. The side lengths do not tell the angle type.
• Remember that \(90^\circ\) is right, not acute and not obtuse.
• An obtuse angle is only a little or a lot bigger than \(90^\circ\), but it is still just one angle opening.
• Focus on the opening between the rays, not where the angle is turned.

Try thinking about these

• \(70^\circ\) is acute because it is less than \(90^\circ\).
• \(90^\circ\) is right because it is exactly \(90^\circ\).
• \(140^\circ\) is obtuse because it is greater than \(90^\circ\).

Quick rule to remember

Think of \(90^\circ\) as the test.

• Less than \(90^\circ\)  acute
• Equal to \(90^\circ\)  right
• Greater than \(90^\circ\)  obtuse

Summary

An angle is made when two rays meet at a vertex. To classify an angle, compare it to a right angle, which is \(90^\circ\).

If the angle is smaller than \(90^\circ\), it is acute. If it is exactly \(90^\circ\), it is right. If it is greater than \(90^\circ\), it is obtuse.

When you are unsure, picture the corner of a square. That can help you decide which kind of angle you see.

Measuring Angles with a Protractor

An angle is made when two lines or rays meet at one point. The point where they meet is called the vertex.

We measure angles to find out how wide they open. Angles are measured in degrees. The degree symbol is written like this: \(^\circ\).

A tool called a protractor helps us measure angles exactly. A protractor is usually shaped like a half-circle and has numbers written from \(0^\circ\) to \(180^\circ\).

Why do protractors have two sets of numbers? One set starts at \(0^\circ\) on the left, and the other set starts at \(0^\circ\) on the right. This helps you measure angles that open in different directions.

Parts of a protractor

• The center point or small hole in the middle
• The straight edge along the bottom
• Number lines that show degrees from \(0^\circ\) to \(180^\circ\)

Steps for measuring an angle

1. Find the vertex of the angle.
2. Place the center point of the protractor exactly on the vertex.
3. Line up one side of the angle with the straight edge at \(0^\circ\).
4. Look at where the other side crosses the numbered edge of the protractor.
5. Read the correct number scale. Start from the \(0^\circ\) that matches the side you lined up.

Important tip: Always begin reading from the side where the angle starts at \(0^\circ\). If you read the wrong set of numbers, your answer will be wrong.

Types of angles you may measure

• Acute angle: less than \(90^\circ\)
• Right angle: exactly \(90^\circ\)
• Obtuse angle: more than \(90^\circ\) but less than \(180^\circ\)
• Straight angle: exactly \(180^\circ\)

You can think of these angle sizes like this:

• If an angle is small, it might be acute.
• If it makes a square corner, it is a right angle.
• If it opens wider than a right angle, it is obtuse.
• If it makes a straight line, it is \(180^\circ\).

Worked Example 1: Measuring a small angle

Suppose one side of the angle is lined up at \(0^\circ\) on the right side of the protractor. The other side crosses the protractor at \(40^\circ\).

The angle measures \(40^\circ\).

Since \(40^\circ < 90^\circ\), it is an acute angle.

Worked Example 2: Measuring a right angle

You place the center point on the vertex. One side is lined up with \(0^\circ\). The other side points to \(90^\circ\).

The angle measures $$90^\circ$$

This is a right angle.

Worked Example 3: Choosing the correct number line

You line up one side of the angle with the left-hand \(0^\circ\). When you look where the other side crosses, you see two numbers: \(120^\circ\) and \(60^\circ\).

Which one do you use?

You must use the scale that starts at the same \(0^\circ\) as your angle side. Since you started at the left-hand \(0^\circ\), the correct measure is \(120^\circ\).

So the angle is \(120^\circ\), which is an obtuse angle.

Worked Example 4: Measuring a straight angle

If the two sides of the angle make one straight line, the protractor reading is \(180^\circ\).

So the angle measure is $$180^\circ$$

This is called a straight angle.

Common mistakes to avoid

• Do not put the edge of the protractor on the vertex. The center point must go on the vertex.
• Do not guess which number to read. Use the scale that begins at the correct \(0^\circ\).
• Do not measure from the wrong side of the angle.
• Make sure one side of the angle is lined up carefully with the protractor's straight edge.

How to check if your answer makes sense

• If the angle looks small, your answer should probably be less than \(90^\circ\).
• If the angle looks like a square corner, your answer should be \(90^\circ\).
• If the angle looks wide, your answer may be more than \(90^\circ\).
• If your answer does not match what the angle looks like, check the number scale again.

Practice thinking

Ask yourself these questions each time:

• Where is the vertex?
• Did I place the center point on the vertex?
• Which side is lined up with \(0^\circ\)?
• Which number line should I read?
• Does my answer match the size of the angle I see?

Summary

To measure an angle with a protractor, place the center on the vertex, line up one side with \(0^\circ\), and read the number where the other side crosses. Remember to use the correct number scale. Angles can be acute, right, obtuse, or straight depending on their degree measure.

Lesson: The Additive Nature of Angles

Have you ever noticed that a big angle can be split into smaller angles? In math, we can add the smaller angles together to find the size of the whole angle. This is called the additive nature of angles.

An angle is formed when two rays meet at one endpoint. The endpoint is called the vertex. Angles are measured in degrees, written with the symbol \(^\circ\).

When one large angle is divided into two or more smaller angles, the smaller angles can be added to find the total. If the smaller angles fit together with no gaps and no overlaps, then their measures make the whole angle.

For example, if one angle is split into angles of \(20^\circ\) and \(35^\circ\), then the whole angle measures:

This means the large angle is \(55^\circ\).

Important idea: This works when the smaller angles are next to each other. Angles that share a side and a vertex are called adjacent angles.

So if two adjacent angles make one larger angle, then:

You can also use subtraction to find a missing part:

Main Teaching Points

• An angle can be broken into smaller angles.
• If the smaller angles are adjacent, you can add them to find the whole angle.
• If you know the whole angle and one part, you can subtract to find the missing part.
• Always check that the smaller angles connect to make the full angle with no gaps or overlaps.

Picture it in your mind: Imagine opening a book a little bit. That opening makes an angle. Now imagine placing a divider inside the opening. The divider splits the big angle into two smaller angles. If you measure both small angles and add them, you get the opening of the whole book.

Worked Example 1: Add Two Small Angles

A large angle is split into two adjacent angles. One angle is \(25^\circ\), and the other angle is \(40^\circ\). What is the measure of the whole angle?

Step 1: Add the two parts.

Answer: The whole angle measures \(65^\circ\).

Worked Example 2: Find a Missing Part

A whole angle measures \(90^\circ\). One smaller angle inside it measures \(30^\circ\). What is the missing angle?

Step 1: Start with the whole angle.

Whole angle = \(90^\circ\)

Step 2: Subtract the known part.

Answer: The missing angle is \(60^\circ\).

Worked Example 3: Add Three Angles

A large angle is split into three adjacent angles: \(15^\circ\), \(20^\circ\), and \(35^\circ\). What is the measure of the whole angle?

Step 1: Add all the parts.

Answer: The whole angle measures \(70^\circ\).

Worked Example 4: A Missing Angle with Three Parts

A whole angle measures \(100^\circ\). Two parts inside it measure \(45^\circ\) and \(25^\circ\). What is the missing third angle?

Step 1: Add the known parts.

Step 2: Subtract from the whole angle.

Answer: The missing angle is \(30^\circ\).

How to Solve These Problems

1. Look for the whole angle and the smaller parts.
2. If all parts are known, add them.
3. If one part is missing, subtract the known part or parts from the whole.
4. Write the degree symbol \(^\circ\) in your answer.

Helpful Tips

• Adjacent angles are side-by-side angles that share the same vertex and one side.
• Only add angles that fit together to make the larger angle.
• If the answer is a missing part, it should be smaller than the whole angle.
• Read carefully to see whether the problem wants the whole or a part.

Try Thinking About These

• If two adjacent angles are \(10^\circ\) and \(50^\circ\), the whole angle is \(60^\circ\).
• If a whole angle is \(80^\circ\) and one part is \(35^\circ\), the missing part is \(45^\circ\).
• If three adjacent angles are \(20^\circ\), \(20^\circ\), and \(20^\circ\), the whole angle is \(60^\circ\).

Summary

The additive nature of angles means that smaller adjacent angles can be added to make a larger angle. If you know the whole angle and one or more parts, you can subtract to find the missing angle. Remember: parts + parts = whole.

Parallel and Perpendicular Lines

In geometry, lines can be placed in different ways. Some lines stay the same distance apart forever. Some lines cross to make square corners. In this lesson, you will learn how to tell the difference between parallel lines and perpendicular lines.

These ideas help us describe shapes, especially polygons like rectangles, squares, and other flat figures. When you can spot parallel and perpendicular lines, you can understand shapes more clearly.

What are parallel lines?

Parallel lines are lines that never meet, even if they keep going forever. They stay the same distance apart all the way along.

You can think of railroad tracks as a real-life example. The tracks run side by side and do not cross. That is what parallel lines do.

Here are some important things to remember about parallel lines:

• They go in the same direction.
• They never touch or cross.
• They stay evenly spaced apart.

We often show parallel lines with matching arrow marks on a drawing.

What are perpendicular lines?

Perpendicular lines are lines that cross to make a right angle. A right angle is a square corner and measures \(90^\circ\).

If two lines meet and make a perfect corner like the corner of a book, a window, or a sheet of paper, those lines are perpendicular.

Here are some important things to remember about perpendicular lines:

• They cross or meet.
• They make a right angle.
• A right angle is written as $$90^\circ$$

How are they different?

• Parallel lines never meet.
• Perpendicular lines do meet.
• Parallel lines stay the same distance apart.
• Perpendicular lines make a square corner when they cross.

Looking for these lines in shapes

Many 2-dimensional shapes have parallel or perpendicular sides.

For example:

• A rectangle has 2 pairs of parallel sides. Its sides also meet to make right angles, so nearby sides are perpendicular.
• A square also has 2 pairs of parallel sides. Its sides meet at right angles too.
• Some other shapes may have only parallel sides, only perpendicular sides, both, or neither.

When you look at a polygon, ask yourself:

1. Do any sides stay the same distance apart and never meet? If yes, they are parallel.
2. Do any sides meet to make a square corner? If yes, they are perpendicular.

Helpful clues

• If two sides are across from each other in a rectangle, they are parallel.
• If two sides touch at a corner of a square or rectangle, they are perpendicular.
• If a corner looks like the corner of a sheet of paper, it is a right angle.

Worked Example 1: Finding parallel lines

A shape has a top side and a bottom side that go straight across. They never meet and stay the same distance apart. Are they parallel or perpendicular?

Step 1: Ask if the sides cross. No, they do not cross.

Step 2: Ask if they stay the same distance apart. Yes, they do.

Answer: The lines are parallel.

Worked Example 2: Finding perpendicular lines

One side goes straight up and down. Another side goes straight left and right. They meet to make a square corner. Are they parallel or perpendicular?

Step 1: Ask if the lines meet. Yes, they do.

Step 2: Ask what kind of angle they make. They make a right angle, or $$90^\circ$$

Answer: The lines are perpendicular.

Worked Example 3: Rectangle sides

Look at a rectangle with sides named top, bottom, left, and right.

• The top and bottom sides are parallel.
• The left and right sides are parallel.
• The top and left sides are perpendicular.
• The top and right sides are perpendicular too.

Why? Opposite sides stay the same distance apart, so they are parallel. Sides that meet at a corner make a right angle, so they are perpendicular.

Worked Example 4: A slanted shape

Imagine a 4-sided shape where the left and right sides lean the same way and never meet. The top and bottom sides also lean the same way and never meet. But none of the corners are square corners.

Step 1: The left and right sides stay the same distance apart, so they are parallel.

Step 2: The top and bottom sides also stay the same distance apart, so they are parallel.

Step 3: Since the corners are not right angles, the sides are not perpendicular.

Answer: This shape has parallel sides, but no perpendicular sides.

Try this way of thinking

When you see two lines, use these questions:

1. Do the lines cross?
2. If they do not cross and stay evenly apart, are they parallel?
3. If they do cross, do they make a right angle?
4. If they make a right angle, are they perpendicular?

Real-world examples

• The opposite edges of a notebook are often parallel.
• The corner of a tile or book shows perpendicular sides.
• Window frames often have both parallel and perpendicular lines.
• Crosswalk stripes are often parallel.

Common mistakes to avoid

• Do not say lines are perpendicular just because they cross. They must cross at a right angle.
• Do not say lines are parallel just because they are near each other. They must stay the same distance apart and never meet.
• Look carefully at corners inside shapes. A square corner means perpendicular sides.

Summary

Parallel lines never meet and stay the same distance apart. Perpendicular lines meet to form a right angle, which is $$90^\circ$$

In shapes like rectangles and squares, opposite sides are parallel, and sides that meet at a corner are perpendicular. When you study a shape, look for sides that never meet and sides that make square corners.

Classifying Polygons by Attributes

In math, a polygon is a closed flat shape made of straight sides. “Closed” means all the sides connect with no openings. “Flat” means it is a 2-dimensional shape, like a shape drawn on paper.

We can classify polygons by attributes. That means we sort and name polygons by looking at their special features, such as:

• how many sides they have,
• how many vertices they have,
• whether sides are equal in length,
• whether sides are parallel,
• whether sides meet at right angles.

A vertex is a corner point where two sides meet. The plural of vertex is vertices.

For any polygon, the number of sides and the number of vertices are the same. For example, a shape with 5 sides also has 5 vertices.

We can write this idea as:

$$\text{number of sides} = \text{number of vertices}$$

Learning to classify polygons helps us describe shapes carefully and correctly.

First, make sure it is a polygon.

A shape is a polygon if it has all of these:

• straight sides,
• the sides close to make a shape,
• no curved parts.

So, a triangle is a polygon. A rectangle is a polygon. But a circle is not a polygon because it has no straight sides.

Classifying by number of sides and vertices

The first and most important way to classify a polygon is by counting its sides or vertices.

• 3 sides and 3 vertices: triangle
• 4 sides and 4 vertices: quadrilateral
• 5 sides and 5 vertices: pentagon
• 6 sides and 6 vertices: hexagon
• 8 sides and 8 vertices: octagon

When you are not sure of a shape’s name, count carefully. Start at one corner and move around the shape until you return to where you started.

Classifying by side lengths

After we count sides, we can look more closely at side lengths.

Some polygons have equal sides. This means two or more sides are the same length.

For example:

• A triangle can have all sides equal, two sides equal, or no equal sides.
• A quadrilateral can also have equal sides, like a square.

Equal side lengths help us describe the shape more exactly.

Classifying by angles

We can also classify polygons by their angles.

A right angle is an angle that forms a square corner. It measures \(90^\circ\).

Shapes with right angles include rectangles and squares.

If a polygon has corners that are not square corners, it may still be a polygon, but it will be a different kind.

Classifying by parallel sides

Parallel sides are sides that stay the same distance apart and never meet.

Some polygons have one pair of parallel sides. Some have two pairs. Some have none.

For example:

• A rectangle has 2 pairs of parallel sides.
• A square has 2 pairs of parallel sides.
• Some quadrilaterals may have only 1 pair of parallel sides.

Classifying by perpendicular sides

Perpendicular sides meet to make a right angle.

If two sides cross or meet like the corner of a square, they are perpendicular.

Rectangles and squares have perpendicular sides because their corners are right angles.

Classifying by symmetry

A shape has symmetry if it can be folded into matching halves.

The fold line is called a line of symmetry.

For example:

• A square has lines of symmetry.
• Some triangles have symmetry, and some do not.
• A rectangle has symmetry too, but not the same number of lines as a square.

Symmetry is another helpful attribute when describing polygons.

Special quadrilaterals

A quadrilateral is any polygon with 4 sides and 4 vertices. There are many kinds of quadrilaterals.

• Rectangle: 4 sides, 4 right angles, and 2 pairs of parallel sides.
• Square: 4 equal sides, 4 right angles, and 2 pairs of parallel sides.
• Rhombus: 4 equal sides.
• Trapezoid: a quadrilateral with 1 pair of parallel sides.

A square is a very special quadrilateral. It belongs to more than one group.

• It is a quadrilateral because it has 4 sides.
• It is also a rectangle because it has 4 right angles.
• It is also a shape with 4 equal sides.

This shows that one polygon can be classified in more than one way.

How to classify a polygon step by step

1. Check that the shape is closed and made of straight sides.
2. Count the sides.
3. Count the vertices to check your work.
4. Look for equal sides.
5. Look for right angles.
6. Look for parallel sides.
7. Look for symmetry.
8. Name the polygon using its attributes.

This step-by-step method makes classifying polygons easier and more accurate.

Worked Example 1

A shape is closed and has 3 straight sides. It has 3 vertices.

Step 1: Is it a polygon? Yes, because it is closed and has straight sides.

Step 2: Count sides and vertices.

It has \(3\) sides and \(3\) vertices.

Step 3: Name the shape.

A polygon with \(3\) sides is a triangle.

Answer: The shape is a triangle.

Worked Example 2

A shape has 4 sides and 4 vertices. Opposite sides are parallel. All 4 angles are right angles.

Step 1: Since it is closed and made of straight sides, it is a polygon.

Step 2: Count sides.

It has \(4\) sides, so it is a quadrilateral.

Step 3: Look at other attributes.

• It has 2 pairs of parallel sides.
• It has 4 right angles.

Step 4: Name the special quadrilateral.

A quadrilateral with 4 right angles is a rectangle.

Answer: The shape is a rectangle.

Worked Example 3

A shape has 4 sides. All 4 sides are equal in length. It also has 4 right angles.

Step 1: It has 4 sides, so it is a quadrilateral.

Step 2: Look at side lengths.

All 4 sides are equal.

Step 3: Look at angles.

All 4 angles are right angles.

Step 4: Name the shape.

A quadrilateral with 4 equal sides and 4 right angles is a square.

Answer: The shape is a square.

Worked Example 4

A shape has 6 straight sides and 6 vertices. It is closed. Not all sides are equal.

Step 1: It is closed and has straight sides, so it is a polygon.

Step 2: Count sides and vertices.

It has \(6\) sides and \(6\) vertices.

Step 3: Name the shape by number of sides.

A polygon with \(6\) sides is a hexagon.

Step 4: Do side lengths change the basic name?

No. Even if the sides are not all equal, it is still a hexagon.

Answer: The shape is a hexagon.

Things to remember

• A polygon must be closed.
• A polygon must have straight sides.
• The number of sides equals the number of vertices.
• Count sides first to find the basic polygon name.
• Then use attributes like equal sides, right angles, parallel sides, perpendicular sides, and symmetry to describe it more clearly.

Quick check questions

• If a polygon has 5 sides, what is it called? Pentagon
• If a shape has 4 sides and 4 right angles, what special quadrilateral could it be? Rectangle
• If a shape has 4 equal sides and 4 right angles, what is it? Square
• Is a circle a polygon? No

Summary

To classify polygons, first check that the shape is closed and made of straight sides. Then count the sides and vertices to name the polygon, such as triangle, quadrilateral, pentagon, or hexagon.

After that, look at other attributes like equal sides, right angles, parallel sides, perpendicular sides, and symmetry. These attributes help you describe the polygon more exactly and tell the difference between shapes like rectangles and squares.

Triangle Classification by Side Length

Triangles are shapes with 3 sides and 3 corners. We can sort, or classify, triangles by looking at their side lengths.

When we classify triangles by side length, we ask: How many sides are the same length?

There are 3 main types of triangles by side length:

• Equilateral triangle
• Isosceles triangle
• Scalene triangle

Let’s learn what each one means.

1. Equilateral Triangle

An equilateral triangle has 3 equal sides.

If the side lengths are the same, then the triangle is equilateral.

Example side lengths:

• \(4, 4, 4\)
• \(7, 7, 7\)

You can think: equi means equal, so equilateral means all sides are equal.

2. Isosceles Triangle

An isosceles triangle has 2 equal sides.

The third side is a different length.

Example side lengths:

• \(5, 5, 3\)
• \(8, 6, 8\)

When you see exactly 2 sides that match, the triangle is isosceles.

3. Scalene Triangle

A scalene triangle has no equal sides.

All 3 side lengths are different.

Example side lengths:

• \(3, 4, 5\)
• \(6, 7, 8\)

If none of the sides match, the triangle is scalene.

How to Classify a Triangle by Its Sides

Follow these steps:

1. Look at the 3 side lengths.
2. Count how many sides are the same.
3. Choose the triangle name:
   • 3 equal sides  equilateral
   • 2 equal sides  isosceles
   • 0 equal sides  scalene

You can organize the idea like this:

Important Note

Be careful to look at side lengths only. In this lesson, we are not naming triangles by their angles. We only check whether the sides are equal or different.

Worked Example 1

Classify the triangle with side lengths \(6, 6, 6\).

Step 1: Look at the side lengths.

They are \(6, 6, 6\).

Step 2: Count how many sides are equal.

All 3 sides are equal.

Answer: This triangle is equilateral.

Worked Example 2

Classify the triangle with side lengths \(9, 4, 9\).

Step 1: Look at the side lengths.

They are \(9, 4, 9\).

Step 2: Count how many sides are equal.

Two sides are equal because \(9 = 9\).

Answer: This triangle is isosceles.

Worked Example 3

Classify the triangle with side lengths \(2, 3, 4\).

Step 1: Look at the side lengths.

They are \(2, 3, 4\).

Step 2: Count how many sides are equal.

No sides are equal.

Answer: This triangle is scalene.

Worked Example 4

A triangle has side lengths \(10\) cm, \(7\) cm, and \(10\) cm. What kind of triangle is it?

Step 1: Compare the side lengths.

The lengths are \(10\), \(7\), and \(10\).

Step 2: Find matching sides.

The first and third sides match because \(10 = 10\).

Step 3: Classify the triangle.

It has exactly 2 equal sides.

Answer: The triangle is isosceles.

Tips to Help You Remember

• Equilateral: all sides equal
• Isosceles: 2 sides equal
• Scalene: all sides different

A quick way to check is to sort the side lengths in your mind and look for matches.

For example:

• \(5, 5, 5\)  equilateral
• \(3, 8, 3\)  isosceles
• \(4, 6, 7\)  scalene

Common Mistakes

• Mistake 1: Counting 2 equal sides in \(6, 6, 6\) and calling it isosceles. Since all 3 sides are equal, it is equilateral.
• Mistake 2: Looking at the shape instead of the side lengths. Always check the lengths carefully.
• Mistake 3: Thinking a triangle with side lengths in a different order changes the name. It does not. For example, \(5, 3, 5\) and \(5, 5, 3\) are both isosceles.

Let’s Review

• A triangle has 3 sides.
• If all 3 sides are equal, it is equilateral.
• If exactly 2 sides are equal, it is isosceles.
• If all 3 sides are different, it is scalene.

Summary

We classify triangles by side length by checking how many sides are the same. An equilateral triangle has 3 equal sides. An isosceles triangle has 2 equal sides. A scalene triangle has no equal sides.

Triangle Classification by Angle Measure

Triangles are shapes with 3 sides and 3 angles. We can sort, or classify, triangles in different ways. In this lesson, we will classify triangles by looking at their angle measures.

The three angle names we will learn are acute, right, and obtuse. To decide which name a triangle gets, we look carefully at its angles.

First, let’s remember the angle types.

• Acute angle: an angle that is less than \(90^\circ\)
• Right angle: an angle that is exactly \(90^\circ\)
• Obtuse angle: an angle that is greater than \(90^\circ\)

You can think of \(90^\circ\) as a square corner. If an angle is smaller than that, it is acute. If it matches that corner exactly, it is right. If it opens wider than that corner, it is obtuse.

Important idea: A triangle is named by its angles using the largest angle.

• If all 3 angles are acute, the triangle is an acute triangle.
• If the triangle has 1 right angle, it is a right triangle.
• If the triangle has 1 obtuse angle, it is an obtuse triangle.

A triangle can only have one right angle or one obtuse angle. If it did not, the shape would not be a triangle.

Also, the angles inside any triangle always add up to:

$$180^\circ$$

This fact helps us check if the angle measures make sense.

1. Acute Triangles

An acute triangle has 3 acute angles. That means every angle is less than \(90^\circ\).

Examples of acute angle measures in a triangle might be:

• \(50^\circ, 60^\circ, 70^\circ\)
• \(45^\circ, 65^\circ, 70^\circ\)

Each angle is less than \(90^\circ\), so these are acute triangles.

2. Right Triangles

A right triangle has 1 right angle. A right angle measures exactly \(90^\circ\).

The other two angles must be acute so that all three angles together still make \(180^\circ\).

Examples of right triangle angle measures are:

• \(90^\circ, 50^\circ, 40^\circ\)
• \(90^\circ, 30^\circ, 60^\circ\)

Because each has one angle that is exactly \(90^\circ\), each one is a right triangle.

3. Obtuse Triangles

An obtuse triangle has 1 obtuse angle. That means one angle is greater than \(90^\circ\).

The other two angles must be acute.

Examples of obtuse triangle angle measures are:

• \(100^\circ, 40^\circ, 40^\circ\)
• \(110^\circ, 30^\circ, 40^\circ\)

Because one angle is greater than \(90^\circ\), each one is an obtuse triangle.

How to Classify a Triangle by Its Angles

1. Look at the angle measures.
2. Find the largest angle.
3. Ask:
   • Is it less than \(90^\circ\)? Then the triangle is acute.
   • Is it exactly \(90^\circ\)? Then the triangle is right.
   • Is it greater than \(90^\circ\)? Then the triangle is obtuse.

This works because the biggest angle tells the triangle’s angle type.

Worked Example 1

A triangle has angles \(55^\circ\), \(65^\circ\), and \(60^\circ\).

Step 1: Find the largest angle. The largest angle is \(65^\circ\).

Step 2: Compare it to \(90^\circ\). Since \(65^\circ < 90^\circ\), it is acute.

Answer: This is an acute triangle.

Worked Example 2

A triangle has angles \(90^\circ\), \(35^\circ\), and \(55^\circ\).

Step 1: Find the largest angle. The largest angle is \(90^\circ\).

Step 2: Compare it to \(90^\circ\). Since it is exactly \(90^\circ\), it is a right angle.

Answer: This is a right triangle.

Worked Example 3

A triangle has angles \(120^\circ\), \(25^\circ\), and \(35^\circ\).

Step 1: Find the largest angle. The largest angle is \(120^\circ\).

Step 2: Compare it to \(90^\circ\). Since \(120^\circ > 90^\circ\), it is obtuse.

Answer: This is an obtuse triangle.

Worked Example 4

A triangle has angles \(80^\circ\), \(10^\circ\), and \(90^\circ\).

Step 1: Find the largest angle. The largest angle is \(90^\circ\).

Step 2: A triangle with one \(90^\circ\) angle is a right triangle.

Step 3: Check the total:

$$80^\circ + 10^\circ + 90^\circ = 180^\circ$$

The measures make a triangle.

Answer: This is a right triangle.

Watch Out for These Mistakes

• Mistake 1: Looking at a smaller angle instead of the largest angle. The largest angle helps name the triangle.
• Mistake 2: Thinking a triangle with one acute angle is an acute triangle. An acute triangle must have all 3 angles acute.
• Mistake 3: Forgetting that a right angle is exactly \(90^\circ\), not less and not more.
• Mistake 4: Forgetting to check whether the angles add to \(180^\circ\).

Quick Check

Try classifying these triangles:

• \(70^\circ, 60^\circ, 50^\circ\)
• \(90^\circ, 45^\circ, 45^\circ\)
• \(100^\circ, 50^\circ, 30^\circ\)

Answers:

• \(70^\circ, 60^\circ, 50^\circ\): acute triangle
• \(90^\circ, 45^\circ, 45^\circ\): right triangle
• \(100^\circ, 50^\circ, 30^\circ\): obtuse triangle

Summary

Triangles can be classified by their angle measures.

• An acute triangle has 3 angles less than \(90^\circ\).
• A right triangle has 1 angle equal to \(90^\circ\).
• An obtuse triangle has 1 angle greater than \(90^\circ\).

To classify a triangle, look at the largest angle. Then compare it to \(90^\circ\). This tells you whether the triangle is acute, right, or obtuse.

Lines of Symmetry in Two-Dimensional Figures

A line of symmetry is a line that divides a shape into two matching halves.

If you could fold the shape on that line and both sides would match exactly, then that line is a line of symmetry.

This is also called reflection symmetry, because one side looks like a mirror image of the other side.

Today, we will learn how to find lines of symmetry in 2-dimensional, or flat, shapes.

What does “matching halves” mean?

Matching halves must be the same size and the same shape. When folded, the edges and corners should line up exactly.

If even one part does not line up, then the line is not a line of symmetry.

Where can a line of symmetry be?

A line of symmetry can go in different directions:

• Vertical — up and down
• Horizontal — side to side
• Diagonal — slanted

A shape can have:

• no lines of symmetry,
• 1 line of symmetry,
• 2 lines of symmetry,
• or even more.

How to check for a line of symmetry

1. Look at the shape carefully.
2. Imagine drawing a line through the shape.
3. Ask: Would both sides match if I folded it there?
4. Check whether the corners and sides would line up exactly.

Important idea: A line of symmetry must split the shape into two equal mirror-image parts. It is not enough for the line to go through the middle. The two sides must also match.

Let’s look at some common shapes.

• A square has 4 lines of symmetry.
• A rectangle has 2 lines of symmetry.
• An equilateral triangle has 3 lines of symmetry.
• A regular hexagon has 6 lines of symmetry.
• A scalene triangle has 0 lines of symmetry.

You do not need to memorize every shape. It is more important to use the folding idea to test each one.

Worked Example 1: Rectangle

How many lines of symmetry does a rectangle have?

A rectangle has opposite sides that are equal. If we draw a vertical line through the center, the left side matches the right side.

If we draw a horizontal line through the center, the top half matches the bottom half.

But the diagonal lines do not make matching halves in a rectangle.

So a rectangle has:

$$2$$

lines of symmetry.

Worked Example 2: Square

How many lines of symmetry does a square have?

A square has all sides equal and all corners equal.

It has:

• 1 vertical line of symmetry
• 1 horizontal line of symmetry
• 2 diagonal lines of symmetry

That makes:

$$4$$

lines of symmetry altogether.

Worked Example 3: Isosceles Triangle

An isosceles triangle has 2 equal sides. Does it have a line of symmetry?

Yes. If the triangle is standing with its point at the top, a line drawn from the top point straight down to the middle of the base makes two matching halves.

This line is vertical.

So an isosceles triangle has:

$$1$$

line of symmetry.

Worked Example 4: Scalene Triangle

A scalene triangle has sides of different lengths. Does it have a line of symmetry?

No. No matter where you draw a line, the two parts will not match exactly.

So a scalene triangle has:

$$0$$

lines of symmetry.

Let’s compare some shapes.

• A shape with more equal sides and equal angles often has more lines of symmetry.
• A shape that is uneven usually has fewer lines of symmetry, or none at all.

Tips for finding lines of symmetry

• Look for a line that goes through the middle of the shape.
• Check both sides carefully.
• Ask whether one side is a mirror image of the other.
• Try vertical, horizontal, and diagonal lines.
• Do not guess just because the shape “looks close.” The halves must match exactly.

Common mistakes

• Thinking every shape has a line of symmetry. Some shapes have none.
• Drawing a line through the middle that does not make matching halves.
• Forgetting to test diagonal lines in shapes like squares.
• Thinking equal area is enough. The two halves must be the same shape too.

Practice thinking

Ask yourself these questions when you look at a shape:

1. Can I fold it so both sides match?
2. Is the fold line vertical, horizontal, or diagonal?
3. How many different fold lines work?

Quick shape review

• Square: 4 lines of symmetry
• Rectangle: 2 lines of symmetry
• Isosceles triangle: 1 line of symmetry
• Scalene triangle: 0 lines of symmetry

Summary

A line of symmetry divides a 2-dimensional figure into two matching mirror-image halves.

To find one, imagine folding the shape along a line. If both sides match exactly, the line is a line of symmetry.

Lines of symmetry can be vertical, horizontal, or diagonal. Some shapes have many lines of symmetry, while some have only one or none.

Rotational symmetry means a shape can be turned, or rotated, and still look exactly the same before it makes one full turn.

A full turn is \(360^\circ\). If a shape matches itself during that turn, then it has rotational symmetry.

Think about spinning a shape around its center. If the shape lands on top of its starting position and all parts line up, then the shape has rotational symmetry.

This is different from line symmetry. Line symmetry uses a fold line. Rotational symmetry uses a turn.

Main idea: We ask, “Can this shape be turned less than \(360^\circ\) and still look exactly the same?” If the answer is yes, the shape has rotational symmetry.

Here are some important words to know:

• Rotate: to turn a shape around a point
• Center: the point a shape turns around
• Full turn: \(360^\circ\)
• Half turn: \(180^\circ\)
• Quarter turn: \(90^\circ\)

Some shapes match themselves after a half turn. Some match after a quarter turn. Some match many times in one full turn. Some do not match at all until \(360^\circ\), and those shapes do not have rotational symmetry.

When checking rotational symmetry, follow these steps:

1. Look at the shape’s center.
2. Imagine turning the shape.
3. See if the corners and sides land exactly on the original shape.
4. If it matches before \(360^\circ\), the shape has rotational symmetry.

Worked Example 1: Square

A square has 4 equal sides and 4 equal corners.

If you turn a square by \(90^\circ\), it looks the same. It also looks the same at \(180^\circ\) and \(270^\circ\).

So a square has rotational symmetry because it matches itself before \(360^\circ\).

The matching turns are:

$$90^\circ,\ 180^\circ,\ 270^\circ$$

Worked Example 2: Rectangle

A rectangle has opposite sides equal. It is longer in one direction than the other unless it is a square.

If you turn a rectangle by \(90^\circ\), it usually does not match, because the long sides and short sides switch places.

But if you turn it by \(180^\circ\), it does match exactly.

So a rectangle has rotational symmetry. It matches after a half turn:

$$180^\circ$$

Worked Example 3: Equilateral Triangle

An equilateral triangle has 3 equal sides and 3 equal angles.

If you turn it by \(120^\circ\), it matches itself. If you turn it by \(240^\circ\), it matches again.

So an equilateral triangle has rotational symmetry.

The matching turns are:

$$120^\circ,\ 240^\circ$$

Worked Example 4: Regular Pentagon

A regular pentagon has 5 equal sides and 5 equal angles.

Because all its sides and corners are arranged evenly, it matches itself several times during a full turn.

To find the turn size, divide the full turn by the number of equal parts:

$$360^\circ \div 5 = 72^\circ$$

So it matches at:

$$72^\circ,\ 144^\circ,\ 216^\circ,\ 288^\circ$$

That means a regular pentagon has rotational symmetry.

A shape that does not have rotational symmetry

Look at a scalene triangle. A scalene triangle has no equal sides.

If you turn it, the sides and corners do not line up with the original shape until the full turn of \(360^\circ\).

Since we only count turns less than \(360^\circ\), a scalene triangle does not have rotational symmetry.

Helpful pattern

Many regular shapes have rotational symmetry. A regular shape has all sides equal and all angles equal.

• A regular triangle matches 2 times before \(360^\circ\).
• A square matches 3 times before \(360^\circ\).
• A regular pentagon matches 4 times before \(360^\circ\).
• A regular hexagon would match 5 times before \(360^\circ\).

The more evenly a shape is built around its center, the more likely it is to have rotational symmetry.

Tips for deciding quickly

• If a shape looks the same after a half turn, it has rotational symmetry.
• If all sides and corners are equal and arranged evenly, it probably has rotational symmetry.
• If one part is different from the others, the shape may not match when turned.
• You must check that all parts line up exactly.

Compare these shapes

• Square: yes, rotational symmetry
• Rectangle: yes, rotational symmetry
• Equilateral triangle: yes, rotational symmetry
• Regular pentagon: yes, rotational symmetry
• Scalene triangle: no rotational symmetry

Remember: The turn must be less than \(360^\circ\). Every shape matches after a full turn, but that does not count.

Let’s practice thinking:

If a shape matches only after \(360^\circ\), then it has no rotational symmetry.

If a shape matches after \(180^\circ\), then it does have rotational symmetry.

If a shape matches after \(90^\circ\), then it also has rotational symmetry.

Brief Summary

Rotational symmetry happens when a shape can be turned around its center and still look exactly the same before making a full \(360^\circ\) turn.

To check, imagine rotating the shape and see whether the sides and corners line up perfectly.

Shapes like squares, rectangles, equilateral triangles, and regular pentagons have rotational symmetry. Shapes that only match after \(360^\circ\) do not.

First Quadrant Coordinate Graphing helps us show where points are on a grid. In 4th grade, you will work in the first quadrant, where all the numbers are positive.

A coordinate grid is made of two number lines that cross.

• The horizontal number line is called the x-axis.
• The vertical number line is called the y-axis.
• They meet at a point called the origin.

The origin is written as $$ (0,0) $$

In the first quadrant, you move:

• right on the x-axis
• up on the y-axis

That means points in the first quadrant look like this: $$ (x,y) $$ where both numbers are 0 or greater.

An ordered pair tells the exact location of a point. It has two numbers in parentheses, like $$ (3,2) $$.

The numbers must be read in the correct order:

1. The first number tells how far to move right.
2. The second number tells how far to move up.

You can remember this with the rule: over, then up.

So for $$ (3,2) $$:

• Move 3 spaces right.
• Then move 2 spaces up.

This lands on the point $$ (3,2) $$.

How to Plot a Point

1. Start at the origin, $$ (0,0) $$.
2. Look at the first number. Move that many spaces right.
3. Look at the second number. Move that many spaces up.
4. Put a dot.
5. Label the point if needed.

How to Name a Point

If a point is already on the grid, you can name it by following its location.

1. Start at the point.
2. Look straight down to the x-axis to find the first number.
3. Look straight across to the y-axis to find the second number.
4. Write the ordered pair as $$ (x,y) $$.

Remember: the x-number comes first, and the y-number comes second.

Worked Example 1: Plot One Point

Plot the point $$ (4,1) $$.

Step 1: Start at $$ (0,0) $$.

Step 2: Move 4 units right.

Step 3: Move 1 unit up.

Answer: Put the point at $$ (4,1) $$.

This point is 4 spaces across and 1 space up.

Worked Example 2: Name a Point

A point is 2 spaces right of the origin and 5 spaces up. What is its ordered pair?

Step 1: Count the spaces right. That is 2.

Step 2: Count the spaces up. That is 5.

Answer: The ordered pair is $$ (2,5) $$.

Be careful not to switch the numbers. $$ (5,2) $$ is a different point.

Worked Example 3: Follow a Path

Sometimes you will plot more than one point and connect them in order. This makes a path.

Plot and connect these points in order:

$$ (1,1), (3,1), (3,4) $$

Step 1: Plot $$ (1,1) $$.

Step 2: Plot $$ (3,1) $$.

Step 3: Plot $$ (3,4) $$.

Step 4: Draw a line segment from $$ (1,1) $$ to $$ (3,1) $$.

Step 5: Draw a line segment from $$ (3,1) $$ to $$ (3,4) $$.

This path goes:

• right from $$ (1,1) $$ to $$ (3,1) $$
• then up from $$ (3,1) $$ to $$ (3,4) $$

This also shows an important idea:

• If the y-number stays the same, the path goes side to side.
• If the x-number stays the same, the path goes up and down.

Worked Example 4: Draw a Simple Shape

Plot and connect these points in order:

$$ (1,1), (4,1), (4,3), (1,3), (1,1) $$

Step 1: Plot each point.

Step 2: Connect them in the order given.

Let's look at the points:

• From $$ (1,1) $$ to $$ (4,1) $$, the y-number stays 1, so the line goes across.
• From $$ (4,1) $$ to $$ (4,3) $$, the x-number stays 4, so the line goes up.
• From $$ (4,3) $$ to $$ (1,3) $$, the y-number stays 3, so the line goes across.
• From $$ (1,3) $$ to $$ (1,1) $$, the x-number stays 1, so the line goes down.

Answer: The points make a rectangle.

Important Things to Remember

• Always start at the origin when plotting.
• x comes first: move right.
• y comes second: move up.
• The order matters. $$ (2,4) $$ and $$ (4,2) $$ are not the same point.
• In the first quadrant, points are on the top-right part of the grid.

Common Mistakes

• Mixing up x and y
  Remember: x is first, y is second.
• Starting from the wrong place
  Always begin at $$ (0,0) $$.
• Counting the lines wrong
  Move carefully one unit at a time.
• Connecting points in the wrong order
  Follow the ordered pairs exactly as given.

Try These on Your Own

1. Plot $$ (2,3) $$.
2. Name the point that is 5 spaces right and 2 spaces up.
3. Plot and connect: $$ (1,2), (4,2), (4,5) $$.
4. Plot and connect: $$ (2,1), (5,1), (5,4), (2,4), (2,1) $$.

As you practice, say the steps out loud: right, then up. This can help you remember how ordered pairs work.

Summary

A coordinate grid helps show where points are. In the first quadrant, you use ordered pairs to move right and up from the origin.

When you plot points and connect them in order, you can make paths and shapes. If you remember x first, y second, you will be able to graph points correctly.