Angles, Lines, and Polygons

Vertical, Adjacent, and Linear Pairs are special relationships between angles. When lines cross or when angles sit next to each other, we can use these relationships to find missing angle measures.

These angle ideas are important because they help us solve geometry problems and algebra problems. If you know how two angles are connected, you can write an equation and solve for an unknown value.

In this lesson, you will learn how to identify adjacent angles, vertical angles, and linear pairs. You will also use the ideas of complementary and supplementary angles to solve problems.

1. Review: What is an angle?

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

Angles are measured in degrees. For example, a right angle measures \(90^\circ\), and a straight angle measures \(180^\circ\).

2. Complementary and Supplementary Angles

Before learning the three main angle relationships, it helps to remember these two important definitions:

• Complementary angles are two angles whose measures add up to \(90^\circ\).
• Supplementary angles are two angles whose measures add up to \(180^\circ\).

These words describe the sum of two angles. The angles do not have to be next to each other unless the problem says so.

For example:

• If one angle is \(35^\circ\), its complement is \(55^\circ\) because \(35 + 55 = 90\).
• If one angle is \(120^\circ\), its supplement is \(60^\circ\) because \(120 + 60 = 180\).

3. Adjacent Angles

Adjacent angles are two angles that are next to each other. They share:

• a common vertex, and
• a common side.

They do not overlap.

If two angles sit side by side, they are adjacent. Adjacent angles can have many different sums. Some adjacent angles add to \(90^\circ\), some add to \(180^\circ\), and some do not make either of those totals.

So, being adjacent tells you about the position of the angles, not their total measure.

4. Vertical Angles

Vertical angles are formed when two lines intersect. They are the opposite angles across from each other.

A very important fact is this:

Vertical angles are always equal.

If two lines cross and one of the angles measures \(70^\circ\), then the angle directly across from it also measures \(70^\circ\).

The other two opposite angles are also equal to each other.

When two lines intersect, four angles are formed:

• one pair of vertical angles, and
• another pair of vertical angles.

5. Linear Pairs

A linear pair is a special kind of adjacent angle pair.

Two angles form a linear pair if:

• they are adjacent, and
• their non-common sides form a straight line.

Because a straight line measures \(180^\circ\), angles in a linear pair are always supplementary.

That means if angles \(A\) and \(B\) form a linear pair, then:

$$A + B = 180$$

So every linear pair is supplementary, but not every supplementary pair is a linear pair. To be a linear pair, the angles must also be adjacent.

6. How these ideas are connected

• Adjacent angles: next to each other.
• Vertical angles: opposite each other when lines intersect; they are equal.
• Linear pair: adjacent angles that form a straight line; they add to \(180^\circ\).
• Complementary angles: add to \(90^\circ\).
• Supplementary angles: add to \(180^\circ\).

7. How to solve angle problems

When solving a problem, ask yourself these questions:

1. Are the angles next to each other? If yes, they may be adjacent.
2. Are the angles opposite each other in intersecting lines? If yes, they are vertical angles.
3. Do the angles form a straight line? If yes, they are a linear pair and sum to \(180^\circ\).
4. Does the problem say complementary or supplementary? Use \(90^\circ\) or \(180^\circ\).
5. Can you write an equation and solve for the variable?

Worked Example 1: Finding a vertical angle

Two lines intersect. One angle measures \(48^\circ\). What is the measure of the vertical angle across from it?

Step 1: Identify the relationship.

The angle across from it is a vertical angle.

Step 2: Use the vertical angle rule.

Vertical angles are equal, so the opposite angle also measures:

$$48^\circ$$

Answer: The vertical angle measures \(48^\circ\).

Worked Example 2: Finding a linear pair

One angle in a linear pair measures \(132^\circ\). Find the measure of the other angle.

Step 1: Use the linear pair rule.

Linear pairs are supplementary, so their measures add to \(180^\circ\).

Let the missing angle be \(x\).

$$x + 132 = 180$$

Step 2: Solve.

$$x = 180 - 132$$

$$x = 48$$

Answer: The other angle measures \(48^\circ\).

Worked Example 3: Solving with algebra using vertical angles

Two vertical angles are labeled \((3x + 5)^\circ\) and \((5x - 19)^\circ\). Find \(x\), then find the angle measure.

Step 1: Use the vertical angle relationship.

Vertical angles are equal, so:

$$3x + 5 = 5x - 19$$

Step 2: Solve the equation.

Subtract \(3x\) from both sides:

$$5 = 2x - 19$$

Add \(19\) to both sides:

$$24 = 2x$$

Divide by \(2\):

$$x = 12$$

Step 3: Find the angle measure.

Substitute \(x = 12\) into one expression:

$$3(12) + 5 = 36 + 5 = 41$$

Answer: \(x = 12\), and each vertical angle measures \(41^\circ\).

Worked Example 4: Solving with algebra using a linear pair

Two angles form a linear pair. Their measures are \((2x + 10)^\circ\) and \((4x - 4)^\circ\). Find \(x\), then find both angle measures.

Step 1: Use the linear pair relationship.

Linear pairs add to \(180^\circ\), so:

$$ (2x + 10) + (4x - 4) = 180 $$

Step 2: Combine like terms.

$$6x + 6 = 180$$

Step 3: Solve.

$$6x = 174$$

$$x = 29$$

Step 4: Find each angle.

First angle:

$$2x + 10 = 2(29) + 10 = 68$$

Second angle:

$$4x - 4 = 4(29) - 4 = 112$$

Check:

$$68 + 112 = 180$$

Answer: \(x = 29\). The angles measure \(68^\circ\) and \(112^\circ\).

8. Common mistakes to avoid

• Do not confuse adjacent and vertical angles. Adjacent angles are side by side. Vertical angles are across from each other.
• Do not assume all adjacent angles are supplementary. They are only supplementary if they form a linear pair or if the problem says so.
• Do not forget the equation. Vertical angles are equal, but linear pairs add to \(180\).
• Check your final answer. If the angles are vertical, they should match. If they are a linear pair, they should add to \(180^\circ\).

9. Quick identification practice

Use these clues:

• If two angles share a side and a vertex, they are adjacent.
• If two angles are opposite when lines cross, they are vertical.
• If two adjacent angles make a straight line, they are a linear pair.

10. Summary

Angle relationships help you understand how angles connect. Adjacent angles are next to each other, vertical angles are opposite and equal, and linear pairs are adjacent angles that add to \(180^\circ\).

When a problem includes variables, use the angle relationship to write an equation. Then solve the equation and check that your answer fits the angle rule.

Parallel Lines and Transversals

When a line crosses two other lines, it can create many angle relationships. In this lesson, you will learn how to recognize and use the angle pairs formed when a transversal crosses parallel lines.

This idea is important because once you know one angle, you can often find several others without measuring. You will also learn the names of special angle pairs: corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

1. Important vocabulary

• Parallel lines are lines in the same plane that never meet. We show parallel lines with the symbol \(\parallel\).
• Transversal means a line that crosses two or more other lines.
• Interior angles are the angles between the two lines.
• Exterior angles are the angles outside the two lines.

Imagine two parallel horizontal lines cut by a slanted line. This slanted line is the transversal. At each crossing, four angles are formed, for a total of eight angles.

2. Why angle relationships happen

When the two lines are parallel, the angle patterns created by the transversal are predictable. Some angle pairs are always equal, and some always add up to \(180^\circ\).

These patterns help us solve problems quickly. Instead of measuring every angle, we use the relationships between the angles.

3. The four main angle pairs

A. Corresponding angles

Corresponding angles are in the same relative position at each intersection.

• For example, one angle might be above the top line and to the right of the transversal.
• Its corresponding angle would be above the bottom line and to the right of the transversal.

If the lines are parallel, then corresponding angles are congruent, which means they have the same measure.

So if angle 1 and angle 5 are corresponding, then

$$m\angle 1 = m\angle 5$$

B. Alternate interior angles

Alternate interior angles are:

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

If the lines are parallel, then alternate interior angles are congruent.

So if angle 3 and angle 6 are alternate interior, then

$$m\angle 3 = m\angle 6$$

C. Alternate exterior angles

Alternate exterior angles are:

• outside the two parallel lines, and
• on opposite sides of the transversal.

If the lines are parallel, then alternate exterior angles are congruent.

So if angle 1 and angle 8 are alternate exterior, then

$$m\angle 1 = m\angle 8$$

D. Consecutive interior angles

Consecutive interior angles are:

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

These angles are supplementary, which means their measures add up to \(180^\circ\).

So if angle 3 and angle 5 are consecutive interior, then

$$m\angle 3 + m\angle 5 = 180^\circ$$

4. A common angle diagram

Many textbooks number the eight angles like this:

At the top intersection, angles \(1, 2, 3, 4\). At the bottom intersection, angles \(5, 6, 7, 8\).

Even if your diagram is labeled differently, the important part is the position of the angles.

• Corresponding: \((1,5), (2,6), (3,7), (4,8)\)
• Alternate interior: \((3,6), (4,5)\)
• Alternate exterior: \((1,8), (2,7)\)
• Consecutive interior: \((3,5), (4,6)\)

5. Other useful angle facts

There are also angle relationships you already know from intersecting lines.

• Vertical angles are opposite each other and are congruent.
• Linear pairs are next to each other on a straight line and add up to \(180^\circ\).

These facts can help you find all the missing angles in a diagram once you know just one angle.

6. How to identify an angle pair

1. Check whether the angle is inside or outside the parallel lines.
2. Check whether the two angles are on the same side or opposite sides of the transversal.
3. Check whether they are in the same relative position at each intersection.

Use these clues:

• Same position \(\rightarrow\) corresponding
• Inside + opposite sides \(\rightarrow\) alternate interior
• Outside + opposite sides \(\rightarrow\) alternate exterior
• Inside + same side \(\rightarrow\) consecutive interior

Worked Example 1: Finding a corresponding angle

Suppose two parallel lines are cut by a transversal, and one angle measures \(65^\circ\). A corresponding angle is asked for.

Because corresponding angles are congruent when lines are parallel, the corresponding angle also measures

$$65^\circ$$

Answer: The corresponding angle is \(65^\circ\).

Worked Example 2: Finding a consecutive interior angle

One consecutive interior angle measures \(112^\circ\). Find the other one.

Consecutive interior angles are supplementary, so they add to \(180^\circ\).

$$x + 112 = 180$$

Subtract \(112\) from both sides:

$$x = 68$$

Answer: The other consecutive interior angle is \(68^\circ\).

Worked Example 3: Solving with an equation

Two alternate interior angles are labeled \((3x + 10)^\circ\) and \((5x - 14)^\circ\). Find \(x\).

Alternate interior angles are congruent, so set them equal:

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

Subtract \(3x\) from both sides:

$$10 = 2x - 14$$

Add \(14\) to both sides:

$$24 = 2x$$

Divide by \(2\):

$$x = 12$$

Now find the angle measure:

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

Answer: \(x = 12\), and each alternate interior angle measures \(46^\circ\).

Worked Example 4: Finding all angles from one angle

In a diagram of parallel lines cut by a transversal, suppose one angle is \(130^\circ\).

Its vertical angle is also \(130^\circ\).

The two angles next to it form linear pairs, so each of those is

$$180 - 130 = 50^\circ$$

Their vertical angles are also \(50^\circ\).

At the other intersection, corresponding, alternate interior, and alternate exterior relationships show that the four angles there will match these same measures: two will be \(130^\circ\) and two will be \(50^\circ\).

Answer: All eight angles are either \(130^\circ\) or \(50^\circ\).

7. Common mistakes to avoid

• Do not guess by how the picture looks. Use the angle positions carefully.
• Make sure the lines are marked as parallel. These special relationships depend on parallel lines.
• Do not confuse alternate interior with consecutive interior. Alternate interior angles are equal, but consecutive interior angles add to \(180^\circ\).
• Remember that corresponding means same relative position, not just “close together.”

8. Quick check

• If two angles are between the lines and on opposite sides of the transversal, they are alternate interior angles.
• If two angles are outside the lines and on opposite sides of the transversal, they are alternate exterior angles.
• If two angles are in the same relative position, they are corresponding angles.
• If two angles are between the lines and on the same side of the transversal, they are consecutive interior angles.

9. Summary

When a transversal crosses parallel lines, special angle relationships are formed. Corresponding angles, alternate interior angles, and alternate exterior angles are congruent. Consecutive interior angles are supplementary, so they add to \(180^\circ\).

If you can identify whether angles are inside or outside the lines, and whether they are on the same side or opposite sides of the transversal, you can name the angle pair and solve for missing angle measures.

Angle Relationships with Parallel Lines

When a line crosses two parallel lines, it creates several angles. These angles are not random. Many of them are connected in special ways, and knowing those relationships helps us find missing angle measures.

In this lesson, you will learn the most important angle relationships formed by a transversal, how to recognize them, and how to use them to solve problems, including algebra problems with variables.

Important vocabulary

• Parallel lines: lines that never meet and stay the same distance apart.
• Transversal: a line that crosses two or more other lines.
• Congruent angles: angles that have the same measure.
• Supplementary angles: two angles whose measures add to \(180^\circ\).

Imagine two parallel lines cut by a transversal. This creates eight angles. Some pairs are equal, and some add to \(180^\circ\).

Main angle relationships

1. Corresponding angles

Corresponding angles are in the same relative position at each intersection. If the lines are parallel, corresponding angles are congruent.

So if one corresponding angle is \(65^\circ\), the other is also \(65^\circ\).

2. Alternate interior angles

These angles are between the parallel lines and on opposite sides of the transversal. If the lines are parallel, alternate interior angles are congruent.

3. Alternate exterior angles

These angles are outside the parallel lines and on opposite sides of the transversal. If the lines are parallel, alternate exterior angles are congruent.

4. Same-side interior angles

These angles are between the parallel lines and on the same side of the transversal. If the lines are parallel, same-side interior angles are supplementary.

That means their measures add to \(180^\circ\).

5. Vertical angles

Vertical angles are opposite each other when two lines intersect. Vertical angles are always congruent, even if the lines are not parallel.

6. Linear pairs

A linear pair is a pair of angles that form a straight line. Linear pairs are always supplementary.

How to solve angle problems with parallel lines

1. Identify the angle relationship.
2. Decide if the angles are equal or if they add to \(180^\circ\).
3. Write an equation.
4. Solve for the variable, if there is one.
5. Substitute the value back in to find the angle measure.

Quick guide

• Corresponding angles \(\rightarrow\) congruent
• Alternate interior angles \(\rightarrow\) congruent
• Alternate exterior angles \(\rightarrow\) congruent
• Same-side interior angles \(\rightarrow\) supplementary
• Vertical angles \(\rightarrow\) congruent
• Linear pair \(\rightarrow\) supplementary

Worked Example 1: Find a missing angle

Two parallel lines are cut by a transversal. One angle measures \(72^\circ\). Find the measure of its alternate interior angle.

Step 1: Identify the relationship.

The angles are alternate interior angles.

Step 2: Use the rule.

Alternate interior angles are congruent when lines are parallel.

Answer:

$$m\angle = 72^\circ$$

Worked Example 2: Use supplementary angles

Two same-side interior angles measure \(110^\circ\) and \(x^\circ\). Find \(x\).

Step 1: Identify the relationship.

Same-side interior angles are supplementary.

Step 2: Write an equation.

$$110 + x = 180$$

Step 3: Solve.

$$x = 180 - 110 = 70$$

Answer:

$$x = 70^\circ$$

Worked Example 3: Solve an algebra problem with congruent angles

Two corresponding angles are labeled \((3x + 15)^\circ\) and \((5x - 25)^\circ\). Find \(x\) and the angle measure.

Step 1: Identify the relationship.

Corresponding angles are congruent.

Step 2: Set the expressions equal.

$$3x + 15 = 5x - 25$$

Step 3: Solve for \(x\).

Subtract \(3x\) from both sides:

$$15 = 2x - 25$$

Add \(25\) to both sides:

$$40 = 2x$$

Divide by \(2\):

$$x = 20$$

Step 4: Find the angle measure.

Substitute \(x = 20\) into either expression.

$$3(20) + 15 = 60 + 15 = 75$$

Check with the other expression:

$$5(20) - 25 = 100 - 25 = 75$$

Answer:

$$x = 20 \quad \text{and} \quad m\angle = 75^\circ$$

Worked Example 4: Multi-step algebra problem

Two same-side interior angles are labeled \((2x + 30)^\circ\) and \((4x - 6)^\circ\). Find \(x\) and both angle measures.

Step 1: Identify the relationship.

Same-side interior angles are supplementary.

Step 2: Write the equation.

$$ (2x + 30) + (4x - 6) = 180 $$

Step 3: Combine like terms.

$$6x + 24 = 180$$

Step 4: Solve for \(x\).

Subtract \(24\) from both sides:

$$6x = 156$$

Divide by \(6\):

$$x = 26$$

Step 5: Find both angle measures.

First angle:

$$2(26) + 30 = 52 + 30 = 82$$

Second angle:

$$4(26) - 6 = 104 - 6 = 98$$

Step 6: Check.

$$82 + 98 = 180$$

Answer:

$$x = 26$$

The angle measures are \(82^\circ\) and \(98^\circ\).

Tips for recognizing angle pairs

• If the angles are in matching corners at each intersection, they are probably corresponding.
• If the angles are inside the parallel lines and across from each other, they are alternate interior.
• If the angles are outside the parallel lines and across from each other, they are alternate exterior.
• If the angles are inside the parallel lines on the same side, they are same-side interior.
• If the angles are opposite at one intersection, they are vertical angles.
• If the angles make a straight line, they are a linear pair.

Common mistakes to avoid

• Do not assume every pair of angles is congruent. Some pairs add to \(180^\circ\) instead.
• Make sure the lines are marked parallel before using these special transversal rules.
• After solving for \(x\), do not forget to plug it back in to find the actual angle measure.
• Check whether the problem asks for the value of \(x\), the angle measure, or both.

Summary

When a transversal cuts parallel lines, special angle relationships appear. Corresponding angles, alternate interior angles, and alternate exterior angles are congruent. Same-side interior angles are supplementary. Vertical angles are congruent, and linear pairs are supplementary.

To solve problems, first name the angle relationship, then write an equation based on whether the angles are equal or add to \(180^\circ\). This helps you solve missing angle problems and multi-step algebra problems with confidence.

Triangle Angle Sum Theorem

Triangles are one of the most important shapes in math. No matter what kind of triangle you draw, its three interior angles always add up to the same total.

This idea is called the Triangle Angle Sum Theorem. It says:

The sum of the interior angles of any triangle is \(180^\circ\).

In math symbols, if a triangle has angles \(A\), \(B\), and \(C\), then:

$$A + B + C = 180^\circ$$

In this lesson, you will learn what this theorem means, why it is true, and how to use it to find missing angles in triangles.

What are interior angles?

The interior angles of a triangle are the angles inside the triangle. Since a triangle has 3 sides, it also has 3 interior angles.

• one angle at each vertex
• all three are inside the triangle
• their total is always \(180^\circ\)

This is true for every triangle:

• small or large
• skinny or wide
• right, acute, or obtuse

Why do the angles add to \(180^\circ\)?

We can give an informal proof using a parallel line and a transversal.

Imagine triangle \(ABC\). We will focus on the top vertex, \(C\). Through point \(C\), draw a line parallel to the base \(AB\).

Now the other two sides of the triangle, \(AC\) and \(BC\), cross that new parallel line. That means they act like transversals.

Because the new line is parallel to \(AB\):

• the angle formed by \(AC\) with the parallel line matches angle \(A\)
• the angle formed by \(BC\) with the parallel line matches angle \(B\)

These matching angles are equal because of alternate interior angles formed by a transversal crossing parallel lines.

At point \(C\), the three angles along the straight line are:

• an angle equal to \(A\)
• the triangle's angle \(C\)
• an angle equal to \(B\)

Angles on a straight line add to \(180^\circ\). So:

$$A + C + B = 180^\circ$$

Rewriting in the usual order:

$$A + B + C = 180^\circ$$

That is why the interior angles of every triangle sum to \(180^\circ\).

Important idea to remember

If you know any two angles in a triangle, you can always find the third angle.

Since all three add to \(180^\circ\), subtract the two known angles from \(180^\circ\).

$$\text{missing angle} = 180^\circ - (\text{angle 1} + \text{angle 2})$$

Types of triangles and angle sums

The theorem works for all types of triangles.

• Acute triangle: all angles are less than \(90^\circ\)
• Right triangle: one angle is exactly \(90^\circ\)
• Obtuse triangle: one angle is greater than \(90^\circ\)

Even though these triangles look different, the angle sum is still \(180^\circ\).

Worked Example 1: Find one missing angle

A triangle has angles \(50^\circ\) and \(60^\circ\). Find the third angle.

Step 1: Write the triangle angle sum equation.

$$50^\circ + 60^\circ + x = 180^\circ$$

Step 2: Add the known angles.

$$110^\circ + x = 180^\circ$$

Step 3: Subtract \(110^\circ\) from \(180^\circ\).

$$x = 70^\circ$$

Answer: The third angle is \(70^\circ\).

Worked Example 2: Right triangle

A right triangle has one angle of \(90^\circ\) and another angle of \(35^\circ\). Find the third angle.

Step 1: Set up the equation.

$$90^\circ + 35^\circ + x = 180^\circ$$

Step 2: Add the known angles.

$$125^\circ + x = 180^\circ$$

Step 3: Subtract.

$$x = 55^\circ$$

Answer: The third angle is \(55^\circ\).

Worked Example 3: Use an expression

In a triangle, two angles measure \(x\) and \(x\), and the third angle measures \(40^\circ\). Find \(x\).

Step 1: Write the angle sum equation.

$$x + x + 40^\circ = 180^\circ$$

Step 2: Combine like terms.

$$2x + 40 = 180$$

Step 3: Subtract \(40\).

$$2x = 140$$

Step 4: Divide by 2.

$$x = 70$$

Answer: Each of those angles is \(70^\circ\).

So the triangle's three angles are \(70^\circ\), \(70^\circ\), and \(40^\circ\).

Worked Example 4: Decide if angle measures form a triangle

Can a triangle have angles \(85^\circ\), \(45^\circ\), and \(60^\circ\)?

Add the angles:

$$85 + 45 + 60 = 190$$

Since \(190^\circ \ne 180^\circ\), these angle measures cannot make a triangle.

Common mistakes to avoid

• Forgetting to use 180 degrees: The interior angles of a triangle always add to \(180^\circ\), not another number.
• Using an exterior angle by mistake: Make sure you are adding the angles inside the triangle.
• Arithmetic errors: Add carefully, then subtract from \(180^\circ\).
• Not checking if the answer makes sense: A triangle angle must be greater than \(0^\circ\).

How this connects to parallel lines

This theorem is closely connected to angle relationships you learned with parallel lines and transversals.

When a line crosses two parallel lines, certain angles are equal, such as alternate interior angles. In the informal proof, drawing a line through one vertex of a triangle parallel to the opposite side lets us match two triangle angles to angles on a straight line.

Since a straight line measures \(180^\circ\), the three triangle angles must also total \(180^\circ\).

Quick strategy for solving problems

1. Write down the three triangle angles.
2. Set their sum equal to \(180^\circ\).
3. Substitute the known values.
4. Solve for the missing angle.
5. Check that all three angles add to \(180^\circ\).

Brief Summary

The Triangle Angle Sum Theorem says that the three interior angles of any triangle always add up to \(180^\circ\).

You can understand this by drawing a line parallel to one side of the triangle and using angle relationships from parallel lines and transversals. This creates angles on a straight line, which add to \(180^\circ\).

If you know two angles in a triangle, you can find the third by subtracting their sum from \(180^\circ\). This theorem is a powerful tool for solving triangle problems and checking whether three angles can form a triangle.

Exterior Angle Theorem is a useful rule about triangles. It helps us find missing angles quickly and understand how angles in a triangle are connected.

In this lesson, you will learn what an exterior angle is, what the theorem says, why it works, and how to use it to solve problems.

First, let’s review two important ideas:

• An interior angle is an angle inside a triangle.
• An exterior angle is formed when one side of a triangle is extended.

Imagine triangle \(ABC\). If you extend one side, the angle on the outside is called an exterior angle.

The two interior angles that are not next to that exterior angle are called the remote interior angles.

Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

In symbols, if \(\angle E\) is an exterior angle of a triangle, and the two remote interior angles are \(\angle 1\) and \(\angle 2\), then

$$m\angle E = m\angle 1 + m\angle 2$$

This means you can add the two far-away interior angles to get the exterior angle.

It also means the exterior angle is always greater than either one of the remote interior angles by itself, because it is their sum.

Why does this theorem work?

We know that the interior angles of a triangle always add up to \(180^\circ\).

$$m\angle A + m\angle B + m\angle C = 180^\circ$$

We also know that an interior angle and its adjacent exterior angle form a straight line, so they add up to \(180^\circ\).

If \(\angle C\) and the exterior angle \(\angle E\) are next to each other, then

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

Since both expressions equal \(180^\circ\), we can set them equal in a useful way. From the triangle sum,

$$m\angle A + m\angle B + m\angle C = 180^\circ$$

And from the straight line,

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

So the outside angle must equal the sum of the other two inside angles:

$$m\angle E = m\angle A + m\angle B$$

That is the Exterior Angle Theorem.

How to use the theorem

1. Find the exterior angle.
2. Identify the two remote interior angles. These are the two interior angles not touching the exterior angle.
3. Add the two remote interior angles.
4. Set that sum equal to the exterior angle.

Worked Example 1: Find the exterior angle

A triangle has remote interior angles of \(50^\circ\) and \(65^\circ\). Find the exterior angle.

Use the Exterior Angle Theorem:

$$m\angle E = 50^\circ + 65^\circ$$ $$m\angle E = 115^\circ$$

Answer: The exterior angle is \(115^\circ\).

Worked Example 2: Find a missing interior angle

An exterior angle of a triangle measures \(130^\circ\). One remote interior angle is \(58^\circ\). Find the other remote interior angle.

Let the missing angle be \(x\).

$$130 = 58 + x$$

Subtract \(58\) from both sides:

$$x = 130 - 58$$ $$x = 72$$

Answer: The missing remote interior angle is \(72^\circ\).

Worked Example 3: Solve an equation with variables

An exterior angle is \((3x + 10)^\circ\). The two remote interior angles are \((x + 20)^\circ\) and \(40^\circ\). Find \(x\).

Use the theorem:

$$3x + 10 = (x + 20) + 40$$

Simplify the right side:

$$3x + 10 = x + 60$$

Subtract \(x\) from both sides:

$$2x + 10 = 60$$

Subtract \(10\):

$$2x = 50$$

Divide by \(2\):

$$x = 25$$

Answer: \(x = 25\).

We can check by finding the angle measures:

• Exterior angle: \(3(25) + 10 = 85^\circ\)
• Remote interior angles: \(25 + 20 = 45^\circ\) and \(40^\circ\)

Since \(45^\circ + 40^\circ = 85^\circ\), the answer is correct.

Worked Example 4: Use the theorem in a multi-step problem

In a triangle, the exterior angle is \(140^\circ\). One interior angle next to it is not needed. The two remote interior angles are \((2x)^\circ\) and \((x + 20)^\circ\). Find \(x\) and then find the two remote interior angles.

Use the Exterior Angle Theorem:

$$140 = 2x + (x + 20)$$

Simplify:

$$140 = 3x + 20$$

Subtract \(20\):

$$120 = 3x$$

Divide by \(3\):

$$x = 40$$

Now find the two angles:

• \(2x = 2(40) = 80^\circ\)
• \(x + 20 = 40 + 20 = 60^\circ\)

Check:

$$80^\circ + 60^\circ = 140^\circ$$

Answer: \(x = 40\), and the remote interior angles are \(80^\circ\) and \(60^\circ\).

Common mistakes to avoid

• Using the wrong two interior angles: Only use the two remote interior angles, not the one next to the exterior angle.
• Adding all three interior angles to the exterior angle: The theorem uses only two interior angles.
• Mixing up interior and exterior angles: The exterior angle is outside the triangle and is formed by extending a side.

Helpful tip

If you see one angle outside a triangle, look across the triangle to the two interior angles that are not touching it. Those are the two angles you add.

Let’s connect this to what you already know

You may already know that the angles inside a triangle add up to \(180^\circ\). The Exterior Angle Theorem is connected to that rule. It gives you a faster way to find an outside angle without first finding the third interior angle.

For example, if the remote interior angles are \(35^\circ\) and \(75^\circ\), then the exterior angle is simply

$$35^\circ + 75^\circ = 110^\circ$$

You do not need to find the inside angle next to the exterior angle first.

Summary

• An exterior angle is formed when one side of a triangle is extended.
• The remote interior angles are the two interior angles not next to that exterior angle.
• The Exterior Angle Theorem says:

$$\text{exterior angle} = \text{sum of the two remote interior angles}$$

This theorem helps you solve for missing angles in triangles, including problems with variables.

Interior and Exterior Angles of Polygons

Polygons are closed shapes made from straight line segments. Examples include triangles, quadrilaterals, pentagons, and hexagons.

In this lesson, you will learn what interior angles and exterior angles are, how to find their sums, and how to find each angle in a regular polygon.

Understanding these angle rules helps you solve many geometry problems quickly and accurately.

1. What is a polygon?

A polygon is a flat, closed figure made of straight sides. The name of a polygon depends on how many sides it has.

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

A polygon with all sides equal and all angles equal is called a regular polygon.

2. Interior angles

An interior angle is an angle inside a polygon. Every vertex of the polygon has one interior angle.

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

Sum of the interior angles

The sum of the interior angles of an n-sided polygon is:

$$ (n-2) \times 180^\circ $$

Here, \(n\) means the number of sides.

Why does this formula work?

You can divide any polygon into triangles by drawing diagonals from one vertex.

A polygon with \(n\) sides can be split into \(n-2\) triangles. Since each triangle has angle sum \(180^\circ\), the total is:

$$ (n-2) \times 180^\circ $$

For example:

• A triangle: \((3-2) \times 180 = 180^\circ\)
• A quadrilateral: \((4-2) \times 180 = 360^\circ\)
• A pentagon: \((5-2) \times 180 = 540^\circ\)
• A hexagon: \((6-2) \times 180 = 720^\circ\)

3. Exterior angles

An exterior angle is formed when one side of a polygon is extended. It is the angle outside the polygon between the extended side and the next side.

Each interior angle and its exterior angle form a linear pair, so they add to \(180^\circ\).

$$ \text{interior angle} + \text{exterior angle} = 180^\circ $$

This means if you know one, you can find the other.

Sum of the exterior angles

The sum of the exterior angles of any polygon, if you take one exterior angle at each vertex going all the way around, is always:

$$ 360^\circ $$

This is true for triangles, quadrilaterals, pentagons, and all other polygons.

4. Regular polygons

In a regular polygon, all sides are equal and all angles are equal. That means:

• All interior angles are the same.
• All exterior angles are the same.

Each exterior angle of a regular polygon

Since the sum of all exterior angles is \(360^\circ\), for a regular polygon with \(n\) sides, each exterior angle is:

$$ \frac{360^\circ}{n} $$

Each interior angle of a regular polygon

Because an interior angle and exterior angle add to \(180^\circ\), each interior angle is:

$$ 180^\circ - \frac{360^\circ}{n} $$

You may also see this written as:

$$ \frac{(n-2) \times 180^\circ}{n} $$

Both formulas give the same answer for the measure of one interior angle in a regular polygon.

5. Important formulas to remember

• Sum of interior angles of an \(n\)-gon: $$ (n-2) \times 180^\circ $$
• Sum of exterior angles of any polygon: $$ 360^\circ $$
• Each exterior angle of a regular \(n\)-gon: $$ \frac{360^\circ}{n} $$
• Each interior angle of a regular \(n\)-gon: $$ 180^\circ - \frac{360^\circ}{n} $$

6. Worked Examples

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

A hexagon has \(n=6\) sides. Use the formula:

$$ (n-2) \times 180^\circ $$ $$ (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ $$

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

Example 2: Find one exterior angle of a regular octagon.

An octagon has \(n=8\) sides. In a regular polygon, each exterior angle is:

$$ \frac{360^\circ}{n} $$ $$ \frac{360^\circ}{8} = 45^\circ $$

Answer: Each exterior angle is \(45^\circ\).

Example 3: Find one interior angle of a regular pentagon.

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

First find each exterior angle:

$$ \frac{360^\circ}{5} = 72^\circ $$

Now subtract from \(180^\circ\):

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

Answer: Each interior angle is \(108^\circ\).

You could also use:

$$ \frac{(5-2) \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ $$

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

Use the regular polygon exterior angle formula:

$$ \frac{360^\circ}{n} = 30^\circ $$

Solve for \(n\):

$$ 360 = 30n $$ $$ n = \frac{360}{30} = 12 $$

Answer: The polygon has 12 sides.

7. Common mistakes to avoid

• Mixing up sum and one angle: \((n-2) \times 180^\circ\) gives the sum of all interior angles, not one interior angle.
• Forgetting “regular”: You can only divide by \(n\) to find one equal angle if the polygon is regular.
• Using the wrong formula: The interior angle sum is not always \(360^\circ\). Only the sum of the exterior angles is always \(360^\circ\).
• Confusing interior and exterior angles: Remember, they are supplementary, so they add to \(180^\circ\).

8. Quick check

1. What is the sum of the interior angles of a pentagon?
2. What is each exterior angle of a regular hexagon?
3. What is each interior angle of a regular decagon?
4. If each exterior angle of a regular polygon is \(40^\circ\), how many sides does it have?

Answers:

1. \((5-2) \times 180 = 540^\circ\)
2. \(360 \div 6 = 60^\circ\)
3. Exterior angle: \(360 \div 10 = 36^\circ\), so interior angle: \(180 - 36 = 144^\circ\)
4. \(360 \div 40 = 9\) sides

9. Summary

Interior angles are the angles inside a polygon, and exterior angles are the angles outside formed by extending a side.

The sum of the interior angles of an \(n\)-sided polygon is $$ (n-2) \times 180^\circ $$. The sum of the exterior angles of any polygon is always $$ 360^\circ $$.

For regular polygons, each exterior angle is $$ \frac{360^\circ}{n} $$ and each interior angle is $$ 180^\circ - \frac{360^\circ}{n} $$. These formulas help you solve many polygon angle problems with confidence.