Geometric Relationships and Transformations

Anatomy of Angles and Lines

Geometry helps us describe shapes, directions, and space. Before we can study larger figures like triangles, quadrilaterals, and transformations, we need to understand the basic building blocks: points, lines, rays, line segments, and angles.

In this lesson, you will learn how to recognize and classify these parts of geometry. You will also learn how to measure angles with a protractor and how to draw them carefully.

1. Points, Lines, Rays, and Line Segments

A point shows an exact location. It has no length or width. We usually name a point with a capital letter, like Point A.

A line is straight and goes on forever in both directions. Because it never ends, a line has no endpoints. A line can be named using two points on it, such as line AB.

A ray starts at one point and goes on forever in one direction. It has one endpoint. For example, ray \(\overrightarrow{AB}\) starts at point \(A\) and passes through point \(B\).

A line segment is part of a line with two endpoints. It has a fixed length. Segment \(\overline{AB}\) starts at \(A\) and ends at \(B\).

• Line: no endpoints, goes forever both ways
• Ray: one endpoint, goes forever one way
• Line segment: two endpoints, fixed length

2. What Is an Angle?

An angle is formed when two rays share the same endpoint. The shared endpoint is called the vertex.

For example, if rays \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\) meet at point \(B\), they form angle \(\angle ABC\). The middle letter tells you the vertex, so the vertex is \(B\).

Angles measure the amount of turning between the two rays. We measure angles in degrees, written with the symbol \(^\circ\).

3. Types of Angles

Angles can be classified by their size.

• Acute angle: greater than \(0^\circ\) and less than \(90^\circ\)
• Right angle: exactly \(90^\circ\)
• Obtuse angle: greater than \(90^\circ\) and less than \(180^\circ\)
• Straight angle: exactly \(180^\circ\)

Here are the angle ranges in a simple way:

4. Naming Angles Correctly

An angle can be named in different ways, but you must be careful. If there is only one angle at a vertex, you may name it by the vertex alone, such as \(\angle B\).

If there is more than one angle at the same vertex, use three letters. The vertex must be the middle letter. For example, \(\angle ABC\) and \(\angle CBD\) are different angles because they use different rays.

5. Measuring Angles with a Protractor

A protractor is a tool used to measure angles. Most protractors are shaped like a half-circle and are marked from \(0^\circ\) to \(180^\circ\).

To measure an angle:

1. Place the center point of the protractor on the vertex of the angle.
2. Line up one ray of the angle with the 0-degree line on the protractor.
3. Look at where the other ray crosses the numbered scale.
4. Read the correct scale. Many protractors have two sets of numbers, so start from the side where your first ray is lined up at \(0^\circ\).

Important tip: If the angle opens to the right, you may use one scale. If it opens to the left, you may need the other scale. Always begin counting from the ray that is lined up with \(0^\circ\).

6. Constructing an Angle with a Protractor

To draw an angle of a given size, follow these steps:

1. Draw one ray with an endpoint.
2. Place the center of the protractor on the endpoint.
3. Line up the ray with \(0^\circ\).
4. Make a mark at the angle measure you want, such as \(50^\circ\) or \(120^\circ\).
5. Use a ruler to draw a second ray from the endpoint through the mark.

This creates an angle with the exact measure you need.

7. Recognizing Relationships Between Lines and Angles

Lines and angles often work together in geometry. Some important ideas are:

• Intersecting lines cross at a point.
• Perpendicular lines intersect to form right angles.
• Parallel lines never meet and stay the same distance apart.
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If two line segments or rays meet and form a \(90^\circ\) angle, they are perpendicular.

If two parts of a line form a straight angle, their measures add to \(180^\circ\).

8. Worked Examples

Example 1: Classify basic figures

Suppose a figure starts at point \(P\) and continues forever through point \(Q\). Is it a line, ray, or line segment?

Step 1: Check the endpoints.

The figure starts at \(P\), so it has one endpoint.

Step 2: See whether it goes on forever.

It continues forever through \(Q\), so it extends in one direction only.

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

Example 2: Classify an angle by its measure

An angle measures \(38^\circ\). What type of angle is it?

Step 1: Compare \(38^\circ\) to the angle categories.

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

Step 2: Since \(38^\circ < 90^\circ\), the angle is acute.

Answer: \(38^\circ\) is an acute angle.

Example 3: Measure an angle on a protractor

A ray is lined up with \(0^\circ\), and the other ray crosses the protractor at \(125^\circ\). What is the angle measure, and how is the angle classified?

Step 1: Read the number where the second ray crosses.

The angle measure is \(125^\circ\).

Step 2: Classify the angle.

Because \(90^\circ < 125^\circ < 180^\circ\), the angle is obtuse.

Answer: The angle measures \(125^\circ\) and is an obtuse angle.

Example 4: Construct an angle

Draw an angle measuring \(70^\circ\).

Step 1: Draw a ray with endpoint \(A\).

Step 2: Place the center of the protractor at \(A\).

Step 3: Line up the ray with \(0^\circ\).

Step 4: Find \(70^\circ\) on the correct scale and make a small mark.

Step 5: Draw a second ray from \(A\) through the mark.

Answer: The new angle is \(70^\circ\), so it is an acute angle.

9. Common Mistakes to Avoid

• Mixing up lines, rays, and segments: Count the endpoints carefully.
• Forgetting the vertex in angle names: In \(\angle ABC\), the vertex is \(B\).
• Reading the wrong protractor scale: Start from the side lined up with \(0^\circ\).
• Guessing angle types by shape only: Measure or compare carefully before classifying.

10. Quick Check

Try these on your own:

1. What is the difference between a ray and a line segment?
2. Is a \(90^\circ\) angle acute, right, obtuse, or straight?
3. What type of angle is \(145^\circ\)?
4. If an angle measures \(180^\circ\), what kind of angle is it?
5. Why is the middle letter important when naming an angle with three letters?

Brief Summary

Points, lines, rays, and line segments are the basic parts of geometry. Angles are formed by two rays with a shared endpoint called the vertex. Angles are measured in degrees and can be classified as acute, right, obtuse, or straight. A protractor helps you measure and draw angles accurately, which is an important skill in geometry.

Angle Pair Relationships help us understand how angles are connected. When two angles are next to each other, across from each other, or add up to a certain total, we can use those relationships to find missing angle measures.

In this lesson, you will learn about four important types of angle pairs:

• Complementary angles
• Supplementary angles
• Vertical angles
• Adjacent angles

You will also learn how to use simple algebra to find unknown angle measures.

First, what is an angle? An angle is formed when two rays meet at one endpoint. Angle measure tells how wide the opening is, and it is measured in degrees.

1. Complementary Angles

Two angles are complementary if their measures add up to .0? wait 90 degrees.

So if angle A and angle B are complementary, then:

$$m\angle A + m\angle B = 90^\circ$$

Example: If one angle measures \(35^\circ\), the other angle is:

$$90^\circ - 35^\circ = 55^\circ$$

So the missing angle is \(55^\circ\).

Complementary angles do not have to be next to each other. They just need to add to \(90^\circ\).

2. Supplementary Angles

Two angles are supplementary if their measures add up to 180 degrees.

If angle C and angle D are supplementary, then:

$$m\angle C + m\angle D = 180^\circ$$

Example: If one angle is \(120^\circ\), the other angle is:

$$180^\circ - 120^\circ = 60^\circ$$

So the missing angle is \(60^\circ\).

Supplementary angles also do not have to be next to each other. They just need to add to \(180^\circ\).

3. Adjacent Angles

Two angles are adjacent if they are side by side. They share:

• a common vertex
• a common side
• no overlapping interior

Adjacent angles are about position, not total measure.

For example, one adjacent angle pair could measure \(20^\circ\) and \(40^\circ\). Another adjacent pair could measure \(90^\circ\) and \(30^\circ\).

Sometimes adjacent angles are also complementary or supplementary, but not always.

4. Vertical Angles

When two lines cross, they form four angles. The angles opposite each other are called vertical angles.

Vertical angles are always equal.

If two angles are vertical angles, then:

$$m\angle 1 = m\angle 2$$

Example: If one vertical angle is \(72^\circ\), the angle across from it is also \(72^\circ\).

The other two angles in the crossing shape would each be supplementary to \(72^\circ\), so they would be:

$$180^\circ - 72^\circ = 108^\circ$$

How to Tell the Difference

• Complementary: add to \(90^\circ\)
• Supplementary: add to \(180^\circ\)
• Adjacent: next to each other
• Vertical: opposite angles formed by intersecting lines

A pair of angles can fit more than one description.

• Two angles can be adjacent and complementary.
• Two angles can be adjacent and supplementary.
• Vertical angles are equal, but they are not adjacent.

Using Algebra with Angle Pairs

Sometimes a missing angle is given as a variable expression, such as \(x + 10\) or \(3x - 5\). You can use the angle relationship to write an equation.

Then solve the equation and substitute the value back in to find the angle measures.

Worked Example 1: Complementary Angles

Two angles are complementary. One angle measures \(28^\circ\). What is the other angle?

Since complementary angles add to \(90^\circ\), write:

$$28 + x = 90$$

Solve for \(x\):

$$x = 90 - 28 = 62$$

Answer: The missing angle is \(62^\circ\).

Worked Example 2: Supplementary Angles with Algebra

Two angles are supplementary. One angle is \((x + 25)^\circ\), and the other is \(95^\circ\). Find \(x\) and both angle measures.

Since supplementary angles add to \(180^\circ\), write:

$$x + 25 + 95 = 180$$

Combine numbers:

$$x + 120 = 180$$

Solve:

$$x = 60$$

Now find the unknown angle:

$$x + 25 = 60 + 25 = 85$$

Answer: \(x = 60\). The two angles are \(85^\circ\) and \(95^\circ\).

Worked Example 3: Vertical Angles

Two vertical angles are labeled \((2x + 10)^\circ\) and \((4x - 30)^\circ\). Find \(x\) and the angle measure.

Vertical angles are equal, so write:

$$2x + 10 = 4x - 30$$

Solve the equation:

$$10 = 2x - 30$$

$$40 = 2x$$

$$x = 20$$

Now substitute \(x = 20\) into either expression:

$$2(20) + 10 = 50$$

Answer: \(x = 20\), and each vertical angle measures \(50^\circ\).

Worked Example 4: Adjacent Supplementary Angles

Two adjacent angles form a straight line. One angle is \((3x)^\circ\) and the other is \((2x + 15)^\circ\). Find \(x\) and both angles.

A straight line measures \(180^\circ\), so the angles are supplementary.

Write the equation:

$$3x + (2x + 15) = 180$$

Combine like terms:

$$5x + 15 = 180$$

Solve:

$$5x = 165$$

$$x = 33$$

Now find each angle:

$$3x = 3(33) = 99$$

$$2x + 15 = 2(33) + 15 = 81$$

Answer: The angles are \(99^\circ\) and \(81^\circ\).

Tips for Solving Angle Pair Problems

1. Read carefully to identify the angle relationship.
2. Ask yourself: Do the angles add to \(90^\circ\), add to \(180^\circ\), or are they equal?
3. Write an equation.
4. Solve the equation.
5. Check that your answer makes sense.

Common Mistakes to Avoid

• Mixing up complementary and supplementary angles
• Thinking adjacent angles must be equal
• Forgetting that vertical angles are opposite, not side by side
• Solving for \(x\) but forgetting to find the actual angle measure

Quick Check

• If two angles add to \(90^\circ\), they are complementary.
• If two angles add to \(180^\circ\), they are supplementary.
• If two angles are side by side, they are adjacent.
• If two angles are opposite when lines cross, they are vertical angles.

Summary

Angle pair relationships help you describe how two angles are connected. Complementary angles add to \(90^\circ\), supplementary angles add to \(180^\circ\), adjacent angles are next to each other, and vertical angles are opposite and equal.

When a problem includes a variable, use the angle relationship to write and solve an equation. Then substitute your value back in to find the missing angle measure.

Transversals and Parallel Lines help us understand how angles are connected when lines cross each other. This is an important idea in geometry because it lets us find missing angle measures and solve for unknown variables.

In this lesson, you will learn what a transversal is, what happens when it crosses parallel lines, and how to identify corresponding angles, alternate interior angles, and same-side interior angles.

By the end, you should be able to look at a diagram, name the angle relationships, and use them to find missing values.

1. What is a transversal?

A transversal is a line that crosses two or more other lines at different points.

If the two lines it crosses are parallel, then special angle relationships are created.

Parallel lines are lines that stay the same distance apart and never meet. We show parallel lines with the symbol \(\parallel\).

For example, if line \(l\) is parallel to line \(m\), we write:

$$l \parallel m$$

2. What angles are formed?

When a transversal crosses two parallel lines, many angles are formed. Some of these angles have special relationships.

• Corresponding angles
• Alternate interior angles
• Same-side interior angles

Let’s understand each one.

3. Corresponding angles

Corresponding angles are in matching positions at each intersection.

If two parallel lines are cut by a transversal, then corresponding angles are equal.

So if one corresponding angle measures \(70^\circ\), the other corresponding angle also measures \(70^\circ\).

Rule:

$$\text{Corresponding angles are congruent}$$

In 7th Grade math, congruent means the angles have the same measure.

4. Alternate interior angles

Interior angles are the angles between the two parallel lines.

Alternate interior angles are:

• between the parallel lines, and
• on opposite sides of the transversal

If two parallel lines are cut by a transversal, alternate interior angles are also equal.

Rule:

$$\text{Alternate interior angles are congruent}$$

For example, if one alternate interior angle is \(115^\circ\), the other alternate interior angle is also \(115^\circ\).

5. Same-side interior angles

Same-side interior angles are:

• between the parallel lines, and
• on the same side of the transversal

These angles are not equal in general. Instead, they add up to \(180^\circ\).

Rule:

$$\text{Same-side interior angles are supplementary}$$

Supplementary means the sum is \(180^\circ\).

For example, if one same-side interior angle is \(60^\circ\), the other is:

$$180^\circ - 60^\circ = 120^\circ$$

6. A quick way to tell them apart

• Corresponding: same position, one at each intersection
• Alternate interior: inside the parallel lines, opposite sides of the transversal
• Same-side interior: inside the parallel lines, same side of the transversal

7. Important facts to remember

• If lines are parallel, corresponding angles are equal.
• If lines are parallel, alternate interior angles are equal.
• If lines are parallel, same-side interior angles add to \(180^\circ\).

These facts help you solve missing angles and equations with variables.

Worked Example 1: Finding a corresponding angle

Two parallel lines are cut by a transversal. One angle measures \(68^\circ\). The angle corresponding to it is \(x\). Find \(x\).

Step 1: Identify the angle relationship.

The problem says the angles are corresponding.

Step 2: Use the rule.

Corresponding angles are equal, so:

$$x = 68$$

Answer: \(x = 68^\circ\)

Worked Example 2: Finding an alternate interior angle

Two parallel lines are cut by a transversal. One alternate interior angle measures \(132^\circ\). The other alternate interior angle is \(y\). Find \(y\).

Step 1: Identify the relationship.

The angles are alternate interior angles.

Step 2: Use the rule.

Alternate interior angles are equal, so:

$$y = 132$$

Answer: \(y = 132^\circ\)

Worked Example 3: Finding a same-side interior angle

Two parallel lines are cut by a transversal. One same-side interior angle is \(47^\circ\). The other is \(z\). Find \(z\).

Step 1: Identify the relationship.

The angles are same-side interior angles.

Step 2: Use the rule.

Same-side interior angles add to \(180^\circ\), so:

$$47 + z = 180$$

Step 3: Solve.

$$z = 180 - 47$$

$$z = 133$$

Answer: \(z = 133^\circ\)

Worked Example 4: Solving for a variable

Two parallel lines are cut by a transversal. Two corresponding angles are labeled \((3x + 10)^\circ\) and \((5x - 14)^\circ\). Find \(x\).

Step 1: Use the angle relationship.

Corresponding angles are equal, so set the expressions equal:

$$3x + 10 = 5x - 14$$

Step 2: Solve the equation.

Subtract \(3x\) from both sides:

$$10 = 2x - 14$$

Add \(14\) to both sides:

$$24 = 2x$$

Divide by \(2\):

$$x = 12$$

Step 3: Check by substituting.

$$3(12) + 10 = 36 + 10 = 46$$

$$5(12) - 14 = 60 - 14 = 46$$

Both expressions give the same angle measure, so the answer is correct.

Answer: \(x = 12\)

8. How to solve problems step by step

1. Look at the diagram carefully.
2. Decide which angle pair is being used.
3. Choose the correct rule:
   • Corresponding \(\rightarrow\) equal
   • Alternate interior \(\rightarrow\) equal
   • Same-side interior \(\rightarrow\) sum of \(180^\circ\)
4. Write an equation.
5. Solve for the missing angle or variable.
6. Check that your answer makes sense.

9. Common mistakes to avoid

• Mixing up angle types: Make sure you check whether the angles are inside the lines or outside, and whether they are on the same side or opposite sides of the transversal.
• Adding when you should set equal: Corresponding and alternate interior angles are equal, not supplementary.
• Setting same-side interior angles equal: Same-side interior angles should add to \(180^\circ\).
• Forgetting that the lines must be parallel: These special rules work because the lines are parallel.

10. Final review

When a transversal crosses two parallel lines, it creates angle pairs with special relationships.

• Corresponding angles are in matching positions and are equal.
• Alternate interior angles are between the lines on opposite sides of the transversal and are equal.
• Same-side interior angles are between the lines on the same side of the transversal and add to \(180^\circ\).

If you can identify the type of angle pair, then you can write the correct equation and solve for the missing measure or variable.

Short Summary

A transversal is a line that crosses two lines. When it crosses parallel lines, special angle relationships appear. Corresponding angles and alternate interior angles are equal, while same-side interior angles add up to \(180^\circ\). These rules help you find missing angles and solve equations with variables.

Triangle Inequality Theorem helps us decide whether three side lengths can make a triangle.

The rule is simple: the sum of any two sides of a triangle must be greater than the third side.

If this is not true, then the sides will not close up to form a triangle.

Imagine trying to build a triangle with three sticks. If one stick is too long compared to the other two, the shorter sticks cannot meet. They will fall flat instead of making a triangle.

For a triangle with side lengths \(a\), \(b\), and \(c\), the Triangle Inequality Theorem says:

$$a+b>c \quad b+c>a \quad a+c>b$$

All three inequalities must be true.

Important idea: the sum must be greater than the third side, not equal to it.

For example, if the side lengths are \(2\), \(3\), and \(5\), then:

$$2+3=5$$

Because the sum is equal to the third side, these lengths do not make a triangle.

How to Check if Side Lengths Form a Triangle

1. Pick any two side lengths and add them.

2. Compare the sum to the third side.

3. Repeat for all three pairs.

4. If every sum is greater than the remaining side, the lengths form a triangle.

Sometimes you can check more quickly by focusing on the longest side.

If the sum of the two shorter sides is greater than the longest side, then the other two inequalities will also work.

Still, when you are learning, it is a good idea to check all three.

Worked Example 1: A Triangle That Works

Do the side lengths \(4\), \(6\), and \(7\) form a triangle?

Check all three sums:

$$4+6=10 \quad \text{and} \quad 10>7$$

$$6+7=13 \quad \text{and} \quad 13>4$$

$$4+7=11 \quad \text{and} \quad 11>6$$

All three inequalities are true.

Answer: Yes, \(4\), \(6\), and \(7\) can form a triangle.

Worked Example 2: A Triangle That Does Not Work

Do the side lengths \(3\), \(5\), and \(9\) form a triangle?

Check the sums:

$$3+5=8 \quad \text{and} \quad 8 \not> 9$$

Right away, one inequality fails.

That means the side lengths cannot form a triangle.

Answer: No, \(3\), \(5\), and \(9\) do not form a triangle.

Worked Example 3: When the Sum Equals the Third Side

Do the side lengths \(2\), \(4\), and \(6\) form a triangle?

Check the sums:

$$2+4=6$$

The sum is equal to the third side, not greater than it.

So these lengths do not make a triangle.

Answer: No, because the Triangle Inequality Theorem says the sum must be greater than the third side.

Worked Example 4: Finding a Missing Side Length

A triangle has side lengths \(5\), \(8\), and \(x\). What values can \(x\) have?

Use the Triangle Inequality Theorem.

First:

$$5+8>x$$

$$13>x$$

So \(x<13\).

Next:

$$5+x>8$$

$$x>3$$

Next:

$$8+x>5$$

$$x>-3$$

This last one is always true if side lengths are positive, so the important limits are:

$$3<x<13$$

Answer: The missing side length must be greater than \(3\) and less than \(13\).

If you are only using whole-number side lengths, possible values are:

$$4, 5, 6, 7, 8, 9, 10, 11, 12$$

Why This Theorem Matters

The Triangle Inequality Theorem helps in many geometry problems.

• It tells whether three lengths can make a triangle.

• It helps find possible values for an unknown side.

• It explains why some shapes cannot close to form triangles.

Common Mistakes to Avoid

• Using equal instead of greater: \(a+b=c\) does not make a triangle.

• Checking only one pair carelessly: make sure the two shorter sides together are longer than the longest side.

• Forgetting side lengths must be positive: triangle sides cannot be \(0\) or negative.

Quick Practice to Think About

Decide whether each set of side lengths can form a triangle:

• \(7, 10, 12\)

• \(1, 2, 4\)

• \(6, 6, 11\)

You can test each one by checking whether the sum of two sides is greater than the third side.

Summary

The Triangle Inequality Theorem says that in any triangle, the sum of any two side lengths must be greater than the third side.

If even one pair adds to less than or equal to the third side, the lengths cannot form a triangle.

You can use this theorem to test side lengths and to find possible values for a missing side.

Interior and Exterior Angles of Polygons

When you look at a polygon, you can study the angles inside the shape and the angles outside the shape. These are called interior angles and exterior angles.

Understanding these angles helps you solve problems about triangles, quadrilaterals, pentagons, hexagons, and many other polygons.

In this lesson, you will learn:

• what interior and exterior angles are,
• how to find the sum of the interior angles of a polygon,
• how to find the sum of the exterior angles of a polygon,
• and how to use these ideas to solve problems.

1. What is a polygon?

A polygon is a closed 2-dimensional shape made of straight sides. Some examples are:

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

The number of sides is often called \(n\).

2. Interior angles

An interior angle is an angle inside a polygon, where two sides meet.

For example, a triangle has 3 interior angles, and a pentagon has 5 interior angles.

Sum of interior angles

For any polygon with \(n\) sides, the sum of the interior angles is:

This is an important formula to remember.

Why does this formula work?

You can divide a polygon into triangles by drawing diagonals from one vertex. Since each triangle has angle sum \(180^\circ\), the total interior angle sum depends on how many triangles are made.

A polygon with \(n\) sides can be divided into \(n-2\) triangles.

So the interior angle sum is:

Examples of interior angle sums

• Triangle: \((3-2)\times 180^\circ = 180^\circ\)
• Quadrilateral: \((4-2)\times 180^\circ = 360^\circ\)
• Pentagon: \((5-2)\times 180^\circ = 540^\circ\)
• Hexagon: \((6-2)\times 180^\circ = 720^\circ\)

3. Exterior angles

An exterior angle is formed when one side of a polygon is extended. The exterior angle is outside the polygon.

At each vertex, the interior angle and the exterior angle next to it make a straight line.

That means:

Sum of exterior angles

For any convex polygon, the sum of one exterior angle at each vertex is always:

This is true no matter how many sides the polygon has.

A convex polygon is a polygon with no corners pushed inward. In 7th Grade, most polygon problems use convex polygons.

Why is the exterior angle sum always \(360^\circ\)?

If you walk all the way around a polygon, you make one full turn. One full turn is \(360^\circ\). That is why the exterior angles add up to \(360^\circ\).

4. Regular polygons

A regular polygon has:

• all sides the same length, and
• all angles the same size.

In a regular polygon, all the interior angles are equal, and all the exterior angles are equal.

Finding one interior angle of a regular polygon

First find the sum of the interior angles:

Then divide by the number of angles, which is \(n\):

Finding one exterior angle of a regular polygon

Since the exterior angles add to \(360^\circ\), divide by the number of sides:

You can also find one interior angle by subtracting the exterior angle from \(180^\circ\):

5. Worked Examples

Example 1: Find the sum of the interior angles of a hexagon.

A hexagon has \(n=6\) sides.

Use the formula:

Substitute \(6\) for \(n\):

Answer: The sum of the interior angles is \(720^\circ\).

Example 2: Find one interior angle of a regular pentagon.

A pentagon has \(n=5\) sides.

First find the sum of the interior angles:

Because the pentagon is regular, all 5 interior angles are equal. Divide by 5:

Answer: One interior angle of a regular pentagon is \(108^\circ\).

Example 3: Find one exterior angle of a regular octagon.

An octagon has \(n=8\) sides.

For a regular polygon:

Substitute \(8\) for \(n\):

Answer: One exterior angle of a regular octagon is \(45^\circ\).

Example 4: A regular polygon has an exterior angle of \(30^\circ\). How many sides does it have?

For a regular polygon:

Set up the equation:

Solve for \(n\):

Answer: The polygon has 12 sides.

6. Important relationships to remember

• Sum of interior angles of an \(n\)-sided polygon: $$ (n-2)\times 180^\circ $$
• Sum of exterior angles of a convex polygon: $$ 360^\circ $$
• Interior angle + exterior angle at the same vertex: $$ 180^\circ $$
• One exterior angle of a regular polygon: $$ \frac{360^\circ}{n} $$

7. Common mistakes to avoid

• Do not use \((n-2)\times 180^\circ\) for exterior angles. That formula is only for the sum of interior angles.
• Do not forget that the exterior angle sum is always \(360^\circ\) for a convex polygon.
• If the polygon is regular, then all the angles are equal. If it is not regular, you cannot assume all angles are the same.
• Be careful to tell whether the question is asking for the sum of the angles or for one angle.

8. Quick check

1. What is the sum of the interior angles of a 7-sided polygon?

   $$ (7-2)\times 180^\circ = 5\times 180^\circ = 900^\circ $$

2. What is the sum of the exterior angles of a convex polygon?

   $$ 360^\circ $$

3. What is one exterior angle of a regular hexagon?

   $$ \frac{360^\circ}{6} = 60^\circ $$

4. What is one interior angle of a regular hexagon?

   $$ 180^\circ - 60^\circ = 120^\circ $$

Summary

Interior angles are the angles inside a polygon. Their sum is found using:

Exterior angles are the angles outside a polygon made by extending a side. For any convex polygon, the sum of one exterior angle at each vertex is always:

In a regular polygon, all angles are equal, so you can divide to find one interior or exterior angle. These formulas make it much easier to solve polygon angle problems.

Quadrilateral Hierarchy is the idea that some quadrilaterals belong to more than one category because they share properties. A quadrilateral is any polygon with 4 sides. In this lesson, you will learn how parallelograms, rectangles, rhombuses, squares, trapezoids, and kites are connected.

Instead of thinking of these shapes as completely separate, we organize them in a hierarchy. A hierarchy shows which groups fit inside other groups. This is similar to a family tree or a Venn diagram.

Understanding this hierarchy helps you classify shapes correctly and explain why a shape belongs in one group, or even several groups, at the same time.

First, remember the main quadrilateral categories.

• Parallelogram: a quadrilateral with both pairs of opposite sides parallel.
• Rectangle: a parallelogram with 4 right angles.
• Rhombus: a parallelogram with 4 equal sides.
• Square: a quadrilateral with 4 equal sides and 4 right angles.
• Trapezoid: a quadrilateral with at least one pair of parallel sides.
• Kite: a quadrilateral with 2 pairs of adjacent equal sides.

Important idea: when mathematicians use a hierarchy, they often use inclusive definitions. That means a shape can belong to a larger group if it has all the required properties.

For example, a rectangle has both pairs of opposite sides parallel. That means every rectangle is also a parallelogram.

Also, a square has 4 right angles, so it is a rectangle. A square also has 4 equal sides, so it is a rhombus. Since rectangles and rhombuses are both parallelograms, every square is also a parallelogram.

This is why the hierarchy matters: a shape can fit in more than one category.

Let’s look more closely at each group.

1. Parallelograms

A parallelogram has:

• 2 pairs of parallel sides
• opposite sides equal in length
• opposite angles equal

Rectangles, rhombuses, and squares are all special kinds of parallelograms.

2. Rectangles

A rectangle has everything a parallelogram has, plus:

• 4 right angles

So, the rectangle group fits inside the parallelogram group.

3. Rhombuses

A rhombus has everything a parallelogram has, plus:

• 4 equal sides

So, the rhombus group also fits inside the parallelogram group.

4. Squares

A square has:

• 4 equal sides
• 4 right angles

That means a square is both a rectangle and a rhombus. Because rectangles and rhombuses are parallelograms, a square is also a parallelogram.

You can think of the square as sitting in the overlap of the rectangle and rhombus groups.

5. Trapezoids

In this lesson, we use the definition: a trapezoid has at least one pair of parallel sides.

Because a parallelogram has 2 pairs of parallel sides, it still has at least one pair. So, using this definition, every parallelogram is also a trapezoid.

This may feel surprising if you learned that trapezoids have exactly one pair of parallel sides. Different textbooks sometimes choose different definitions. For a hierarchy and Venn diagram, the inclusive definition at least one pair works best because it shows how groups are related.

6. Kites

A kite has:

• 2 pairs of adjacent equal sides

Adjacent means next to each other. So if one side is equal to the side beside it, and another side is equal to the side beside it, the shape may be a kite.

Some rhombuses and squares also fit the kite definition because all 4 sides are equal. That means you can form 2 pairs of adjacent equal sides.

So in an inclusive hierarchy, some kites overlap with rhombuses and squares.

How to use a hierarchy or Venn diagram

When placing shapes in a Venn diagram, ask these questions:

1. Does it have at least one pair of parallel sides?
2. Does it have two pairs of parallel sides?
3. Does it have 4 right angles?
4. Does it have 4 equal sides?
5. Does it have 2 pairs of adjacent equal sides?

Each “yes” gives you more information about where the shape belongs.

A simple way to picture the hierarchy

• All of these shapes are quadrilaterals.
• Inside quadrilaterals, there is the group of trapezoids.
• Inside trapezoids, there is the group of parallelograms.
• Inside parallelograms, there are rectangles and rhombuses.
• Where rectangles and rhombuses overlap, you get squares.
• Kites form another group that overlaps with rhombuses and squares.

Worked Example 1: Is every rectangle a parallelogram?

Question: Decide whether the statement is true or false: “Every rectangle is a parallelogram.”

Step 1: Recall the definition of a parallelogram: both pairs of opposite sides are parallel.

Step 2: A rectangle has 4 right angles. Because of its shape, its opposite sides are parallel.

Conclusion: The statement is true.

Answer: Every rectangle is a parallelogram.

Worked Example 2: Is every rhombus a square?

Question: Decide whether the statement is true or false: “Every rhombus is a square.”

Step 1: A rhombus must have 4 equal sides.

Step 2: A square must have 4 equal sides and 4 right angles.

Step 3: A rhombus does not always have 4 right angles.

Conclusion: The statement is false.

Answer: Some rhombuses are squares, but not all rhombuses are squares.

Worked Example 3: Classify a shape with properties

Question: A quadrilateral has 4 equal sides and 4 right angles. Name every category it belongs to.

Step 1: 4 equal sides means it is a rhombus.

Step 2: 4 right angles means it is a rectangle.

Step 3: A shape that is both a rhombus and a rectangle is a square.

Step 4: Since squares are also parallelograms, it is a parallelogram.

Step 5: Since parallelograms have at least one pair of parallel sides, it is also a trapezoid.

Step 6: Since all 4 sides are equal, it also has 2 pairs of adjacent equal sides, so it is a kite.

Answer: The shape is a square, rectangle, rhombus, parallelogram, trapezoid, kite, and of course a quadrilateral.

Worked Example 4: Use angle and side information

Question: A quadrilateral has both pairs of opposite sides parallel, but it does not have 4 right angles and not all 4 sides are equal. How should it be classified?

Step 1: Both pairs of opposite sides parallel means it is a parallelogram.

Step 2: It does not have 4 right angles, so it is not a rectangle.

Step 3: Not all 4 sides are equal, so it is not a rhombus.

Step 4: If it is not both a rectangle and a rhombus, then it is not a square.

Step 5: Since it has at least one pair of parallel sides, it is also a trapezoid.

Answer: It is a parallelogram, a trapezoid, and a quadrilateral.

Common mistakes to avoid

• Mistake 1: Thinking categories never overlap. In a hierarchy, they often do.
• Mistake 2: Thinking a square is only a square. A square also belongs to other groups.
• Mistake 3: Forgetting the definition being used for trapezoid. In this lesson, it means at least one pair of parallel sides.
• Mistake 4: Mixing up opposite sides and adjacent sides.

Quick property chart

• Parallelogram: 2 pairs of parallel sides
• Rectangle: parallelogram + 4 right angles
• Rhombus: parallelogram + 4 equal sides
• Square: rectangle + rhombus
• Trapezoid: at least 1 pair of parallel sides
• Kite: 2 pairs of adjacent equal sides

One more helpful way to think about it

If a shape has more special properties, it can belong to a smaller, more specific group.

For example:

• A parallelogram is a broad group.
• A rectangle is more specific because it needs 4 right angles.
• A square is even more specific because it needs 4 right angles and 4 equal sides.

So the hierarchy moves from general groups to special groups.

Summary

A quadrilateral hierarchy shows how 4-sided shapes are related. Rectangles, rhombuses, and squares are special kinds of parallelograms. Squares belong to both the rectangle and rhombus groups. Using the inclusive definition, parallelograms are also trapezoids, and some kites overlap with rhombuses and squares. When classifying a quadrilateral, always look carefully at its sides, angles, and parallel lines.

Pythagorean Theorem Foundations

The Pythagorean Theorem is a rule that helps us find missing side lengths in a right triangle. A right triangle is a triangle that has one angle measuring 90°.

This theorem is one of the most useful ideas in geometry because it connects the lengths of the three sides of a right triangle. If you know any two side lengths, you can find the third.

Before using the theorem, it is important to know the names of the sides of a right triangle.

• Legs: the two sides that form the right angle
• Hypotenuse: the side across from the right angle; it is always the longest side

We usually label the two legs as \(a\) and \(b\), and the hypotenuse as \(c\).

The Pythagorean Theorem says:

$$a^2 + b^2 = c^2$$

This means that if you square the length of one leg, square the length of the other leg, and add those two results, you get the square of the hypotenuse.

Important: This rule only works for right triangles.

Let’s make sense of the equation:

• \(a^2\) means \(a \times a\)
• \(b^2\) means \(b \times b\)
• \(c^2\) means \(c \times c\)

For example, if a side length is 4, then:

$$4^2 = 4 \times 4 = 16$$

If a side length is 7, then:

$$7^2 = 7 \times 7 = 49$$

To use the theorem correctly, follow these steps:

1. Check that the triangle is a right triangle.
2. Identify the hypotenuse. It is opposite the 90° angle.
3. Label the legs as \(a\) and \(b\), and the hypotenuse as \(c\).
4. Substitute the known values into \(a^2 + b^2 = c^2\).
5. Solve for the missing side.

Worked Example 1: Find the hypotenuse

A right triangle has legs of length 3 and 4. Find the hypotenuse.

Step 1: Write the formula.

$$a^2 + b^2 = c^2$$

Step 2: Substitute the side lengths.

$$3^2 + 4^2 = c^2$$

Step 3: Square the numbers.

$$9 + 16 = c^2$$

Step 4: Add.

$$25 = c^2$$

Step 5: Find the number whose square is 25.

$$c = 5$$

So, the hypotenuse is 5.

Worked Example 2: Find a missing leg

A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

Step 1: Use the formula.

$$a^2 + b^2 = c^2$$

Let the missing leg be \(b\). Then \(a = 5\) and \(c = 13\).

Step 2: Substitute.

$$5^2 + b^2 = 13^2$$

Step 3: Square the known numbers.

$$25 + b^2 = 169$$

Step 4: Subtract 25 from both sides.

$$b^2 = 144$$

Step 5: Find the square root of 144.

$$b = 12$$

So, the missing leg is 12.

Worked Example 3: Distance on a grid

Suppose you move 6 units to the right and 8 units up on a coordinate grid. The straight-line distance between the start and end points forms the hypotenuse of a right triangle. Find that distance.

The horizontal distance and vertical distance are the legs, so:

$$6^2 + 8^2 = c^2$$

$$36 + 64 = c^2$$

$$100 = c^2$$

$$c = 10$$

The straight-line distance is 10 units.

This is one way the Pythagorean Theorem helps us find missing two-dimensional distances.

Worked Example 4: Is it a right triangle?

A triangle has side lengths 7, 24, and 25. Is it a right triangle?

To check, use the longest side as \(c\). Here, \(25\) is the longest side.

Now test the equation:

$$7^2 + 24^2 = 25^2$$

$$49 + 576 = 625$$

$$625 = 625$$

The equation is true, so this is a right triangle.

Common Mistakes to Avoid

• Do not use the theorem unless the triangle is a right triangle.
• Do not confuse a leg with the hypotenuse.
• Remember that the hypotenuse is always the longest side.
• Square first, then add or subtract.
• If you are finding a leg, you will usually need to subtract after squaring.

Helpful Pattern

Some right triangles have side lengths that appear often. These are useful to recognize:

• \(3, 4, 5\)
• \(5, 12, 13\)
• \(7, 24, 25\)

If you see these numbers, they can help you solve problems faster.

Why the Theorem Works

The theorem shows a special relationship between the three sides of a right triangle. The total of the squares on the two legs is equal to the square on the hypotenuse.

You can think of it as comparing areas of squares built on each side of the triangle. The two smaller square areas together equal the area of the largest square.

For example, in the \(3,4,5\) triangle:

$$3^2 + 4^2 = 5^2$$

$$9 + 16 = 25$$

The areas of the two smaller squares, \(9\) and \(16\), add to make \(25\), the area of the largest square.

Summary

The Pythagorean Theorem is used with right triangles and states that:

$$a^2 + b^2 = c^2$$

The legs are the two sides that meet at the right angle, and the hypotenuse is the longest side across from the right angle.

You can use this theorem to:

• find the hypotenuse when you know both legs
• find a missing leg when you know the hypotenuse and one leg
• find straight-line distance in two dimensions
• check whether a triangle is a right triangle

When solving, always identify the hypotenuse first and make sure the triangle is a right triangle.

Rigid transformations are movements of a figure that do not change its size or shape. After a rigid transformation, the new figure is congruent to the original figure. That means the figures have the same side lengths and the same angle measures.

In 7th grade, the three main rigid transformations are translations, reflections, and rotations. You can think of them as a slide, a flip, and a turn.

These transformations are often shown on the coordinate plane. A point such as \((2,3)\) can move to a new location depending on the transformation. The moved figure is often called the image, and the original figure is called the preimage.

For example, if point \(A\) is the original point, its image might be written as \(A'\) (read “A prime”).

Why rigid transformations matter:

• They help us describe how shapes move.
• They show when two figures are congruent.
• They help us understand symmetry and patterns.
• They are a big part of geometry on the coordinate plane.

Important idea: A rigid transformation preserves:

• side lengths
• angle measures
• shape
• size

Only the position or orientation may change.

1. Translation (slide)

A translation moves every point of a figure the same distance in the same direction. Nothing turns and nothing flips.

If a figure moves 4 units right and 2 units up, then every vertex does the same thing.

On a coordinate plane:

• moving right means adding to the \(x\)-coordinate
• moving left means subtracting from the \(x\)-coordinate
• moving up means adding to the \(y\)-coordinate
• moving down means subtracting from the \(y\)-coordinate

If a point \((x,y)\) is translated right \(a\) units and up \(b\) units, the image is:

$$ (x,y) \rightarrow (x+a, y+b) $$

Example: Translate \((3,-1)\) left 5 units and up 2 units.

Left 5 means subtract 5 from \(x\). Up 2 means add 2 to \(y\).

$$ (3,-1) \rightarrow (3-5, -1+2) = (-2,1) $$

2. Reflection (flip)

A reflection flips a figure across a line. The line is called the line of reflection. Each point and its image are the same distance from the line of reflection, but on opposite sides.

Common reflection lines are the x-axis and the y-axis.

Reflection across the x-axis:

The \(x\)-coordinate stays the same, and the \(y\)-coordinate changes sign.

$$ (x,y) \rightarrow (x,-y) $$

Reflection across the y-axis:

The \(y\)-coordinate stays the same, and the \(x\)-coordinate changes sign.

$$ (x,y) \rightarrow (-x,y) $$

Examples:

• Across the x-axis: \((4,2) \rightarrow (4,-2)\)
• Across the y-axis: \((4,2) \rightarrow (-4,2)\)

A reflection changes the figure’s orientation. It looks like a mirror image.

3. Rotation (turn)

A rotation turns a figure around a fixed point called the center of rotation. In many 7th grade problems, the center is the origin, which is \((0,0)\).

The most common rotations are 90°, 180°, and 270°. A rotation can be clockwise or counterclockwise.

When rotating around the origin, these rules are helpful:

• 90° counterclockwise: $$ (x,y) \rightarrow (-y,x) $$
• 180°: $$ (x,y) \rightarrow (-x,-y) $$
• 90° clockwise: $$ (x,y) \rightarrow (y,-x) $$

You do not need to memorize too many rules at once. Focus on understanding that the point turns around the origin while staying the same distance from it.

Example: Rotate \((2,5)\) 90° clockwise around the origin.

Using the rule:

$$ (x,y) \rightarrow (y,-x) $$

$$ (2,5) \rightarrow (5,-2) $$

How to tell transformations apart

• Translation: the figure slides; same direction for every point
• Reflection: the figure flips across a line
• Rotation: the figure turns around a point

What stays the same in all rigid transformations?

• side lengths
• angle measures
• area
• perimeter

What may change?

• location on the plane
• direction the figure faces

Worked Example 1: Translation of a triangle

Triangle \(ABC\) has vertices \(A(1,2)\), \(B(4,2)\), and \(C(2,5)\). Translate the triangle 3 units right and 4 units down.

Step 1: Write the translation rule.

Right 3 means add 3 to \(x\). Down 4 means subtract 4 from \(y\).

$$ (x,y) \rightarrow (x+3,y-4) $$

Step 2: Move each vertex.

$$ A(1,2) \rightarrow A'(1+3,2-4) = (4,-2) $$

$$ B(4,2) \rightarrow B'(4+3,2-4) = (7,-2) $$

$$ C(2,5) \rightarrow C'(2+3,5-4) = (5,1) $$

Answer: The image is triangle \(A'B'C'\) with vertices \(A'(4,-2)\), \(B'(7,-2)\), and \(C'(5,1)\).

Worked Example 2: Reflection across an axis

Point \(P(-6,3)\) is reflected across the y-axis. What is the image?

Step 1: Use the reflection rule.

Across the y-axis:

$$ (x,y) \rightarrow (-x,y) $$

Step 2: Apply the rule.

$$ (-6,3) \rightarrow (6,3) $$

Answer: \(P'(6,3)\)

Check your thinking: The point was 6 units left of the y-axis. After reflecting, it is 6 units right of the y-axis. The height stays the same.

Worked Example 3: Rotation around the origin

Quadrilateral \(JKLM\) has one vertex at \(J(3,1)\). Find the image of \(J\) after a 180° rotation around the origin.

Step 1: Use the 180° rule.

$$ (x,y) \rightarrow (-x,-y) $$

Step 2: Apply the rule.

$$ (3,1) \rightarrow (-3,-1) $$

Answer: \(J'(-3,-1)\)

What this means: A 180° rotation sends the point to the opposite side of the origin.

Worked Example 4: Identify the transformation

A point moves from \((2,4)\) to \((-2,4)\). What rigid transformation could this be?

The \(x\)-coordinate changed sign, but the \(y\)-coordinate stayed the same.

That matches reflection across the y-axis.

$$ (x,y) \rightarrow (-x,y) $$

Answer: The transformation is a reflection across the y-axis.

Tips for graphing rigid transformations

1. Plot the original points carefully.
2. Apply the rule to each vertex one at a time.
3. Label the image points with prime marks, such as \(A'\), \(B'\), and \(C'\).
4. Check that the figure kept the same size and shape.

Common mistakes to avoid

• Translation: moving some points different amounts. Every point must move the same way.
• Reflection: mixing up the x-axis and y-axis rules.
• Rotation: confusing clockwise and counterclockwise.
• Forgetting that rigid transformations keep the figure congruent.

Quick comparison chart

• Translation: slide; no turning or flipping
• Reflection: flip over a line
• Rotation: turn around a point

How rigid transformations connect to congruence

If one figure can be moved onto another figure using translations, reflections, and rotations, then the figures are congruent.

That is because rigid transformations do not stretch or shrink figures. They only move them.

Summary

Rigid transformations are movements that keep a figure’s size and shape the same. The three main types are translations (slides), reflections (flips), and rotations (turns).

On the coordinate plane, you can describe these transformations using rules for how the coordinates change. When you understand how points move, you can graph the image of any figure and decide whether figures are congruent.

Dilations and Similarity

In geometry, we often move shapes in different ways. Some moves, like sliding, flipping, and turning, keep the shape exactly the same size. A dilation is different. A dilation changes the size of a figure, but it keeps the shape the same.

When a figure is enlarged or reduced but still has the same shape, the original figure and the new figure are called similar. In this lesson, you will learn what dilations are, how to use a scale factor, and how dilations create similar figures on the coordinate plane.

1. What is a dilation?

A dilation is a transformation that makes a figure larger or smaller. Every point of the figure moves away from or toward a fixed point called the center of dilation.

• If the figure gets larger, it is an enlargement.
• If the figure gets smaller, it is a reduction.

The amount of resizing is described by the scale factor.

• If the scale factor is greater than 1, the image gets larger.
• If the scale factor is between 0 and 1, the image gets smaller.
• If the scale factor is 1, the figure stays the same size.

For example:

• Scale factor of 2 means every length becomes twice as long.
• Scale factor of \(\frac{1}{2}\) means every length becomes half as long.

2. Important words

• Preimage: the original figure
• Image: the new figure after the transformation
• Center of dilation: the fixed point from which the figure grows or shrinks
• Scale factor: the number that tells how much to enlarge or reduce

3. What happens in a dilation?

A dilation changes the size of side lengths, but it does not change the shape. This means:

• Corresponding side lengths are multiplied by the scale factor.
• Corresponding angles stay the same.
• The image and the preimage are similar figures.

So if a triangle has side lengths 3, 4, and 5, and it is dilated by a scale factor of 2, the new side lengths are:

The angle measures do not change.

4. Dilations on the coordinate plane

When the center of dilation is the origin, which is \((0,0)\), there is a simple rule for finding the image of a point.

If the original point is \((x,y)\) and the scale factor is \(k\), then the image is:

This means you multiply both coordinates by the scale factor.

For example:

• If \((3,4)\) is dilated by a scale factor of 2, the image is \((6,8)\).
• If \((6,-2)\) is dilated by a scale factor of \(\frac{1}{2}\), the image is \((3,-1)\).

5. Worked Example 1: Dilating one point

Point \(A(2,5)\) is dilated from the origin with a scale factor of 3. Find the image of the point.

Step 1: Use the rule \((x,y) \to (kx,ky)\).

Step 2: Multiply each coordinate by 3.

Answer: The image is \(A'(6,15)\).

6. Worked Example 2: Dilating a triangle

Triangle \(ABC\) has vertices:

• \(A(1,1)\)
• \(B(4,1)\)
• \(C(1,3)\)

Dilate the triangle from the origin using a scale factor of 2.

Step 1: Multiply each coordinate by 2.

Answer: The image has vertices:

• \(A'(2,2)\)
• \(B'(8,2)\)
• \(C'(2,6)\)

This new triangle is larger than the original, so it is an enlargement. The new triangle has the same angle measures as the original, so the triangles are similar.

7. Worked Example 3: Reduction with a fraction scale factor

Point \(P(8,-6)\) is dilated from the origin by a scale factor of \(\frac{1}{2}\). Find the image.

Step 1: Multiply each coordinate by \(\frac{1}{2}\).

Answer: The image is \(P'(4,-3)\).

Because the scale factor is less than 1, the image is smaller than the original. This is a reduction.

8. How do we know figures are similar?

Two figures are similar if:

• their corresponding angles are equal, and
• their corresponding side lengths have the same scale factor.

A dilation creates a similar figure because it multiplies all lengths by the same number.

Example: A rectangle has side lengths 4 and 6. Another rectangle has side lengths 8 and 12.

Check the scale factor:

Both side lengths were multiplied by 2, so the rectangles are similar.

9. Worked Example 4: Deciding if figures are similar

Triangle 1 has side lengths 3, 5, and 7.

Triangle 2 has side lengths 6, 10, and 14.

Are the triangles similar?

Step 1: Compare corresponding side lengths.

Step 2: Check whether all side lengths use the same scale factor.

Yes, all the side lengths are multiplied by 2.

Answer: The triangles are similar.

Now look at a different example.

Triangle 1 has side lengths 2, 3, and 4.

Triangle 2 has side lengths 4, 6, and 7.

Compare the side lengths:

Because the scale factor is not the same for all side lengths, these triangles are not similar.

10. What stays the same and what changes?

It is helpful to remember which parts of a figure stay the same during a dilation and which parts change.

• Stays the same: angle measures, shape
• Changes: side lengths, perimeter, distance from the center of dilation

If the scale factor is 3, then each side length becomes 3 times as long. If the original perimeter was 10 units, the new perimeter becomes:

11. Common mistakes to avoid

• Forgetting to multiply both coordinates: In a dilation from the origin, multiply both \(x\) and \(y\) by the scale factor.
• Mixing up enlargement and reduction: A scale factor greater than 1 enlarges. A scale factor between 0 and 1 reduces.
• Thinking congruent and similar mean the same thing: Congruent figures are the same shape and same size. Similar figures are the same shape but may be different sizes.
• Using different scale factors for different sides: For figures to be similar, all corresponding side lengths must have the same scale factor.

12. Quick practice questions

1. Point \((3,2)\) is dilated from the origin by a scale factor of 4. What is the image?
2. Point \((-8,6)\) is dilated from the origin by a scale factor of \(\frac{1}{2}\). What is the image?
3. A triangle has side lengths 4, 6, and 8. After dilation, the side lengths are 10, 15, and 20. Are the triangles similar?
4. A square has side length 5. It is dilated by a scale factor of 3. What is the new side length?

Answers:

1. \((12,8)\)
2. \((-4,3)\)
3. Yes, because each side is multiplied by \(\frac{5}{2}\).
4. 15

Summary

A dilation changes the size of a figure using a scale factor and a center of dilation. When the center is the origin, you can find the image of a point by multiplying each coordinate by the scale factor: \((x,y) \to (kx,ky)\).

Dilations create similar figures. Similar figures have the same shape, equal corresponding angles, and side lengths that are all multiplied by the same number. If you remember how scale factors work, you can resize figures and check whether two figures are similar.