# 3.4: Alternate Interior Angles

**At Grade**Created by: CK-12

**Practice**Alternate Interior Angles

What if you were presented with two angles that are on opposite sides of a transversal, but inside the lines? How would you describe these angles and what could you conclude about their measures? After completing this Concept, you'll be able to answer these questions using your knowledge of alternate interior angles.

### Watch This

CK-12 Foundation: Chapter3AlternateInteriorAnglesA

Watch the portions of this video dealing with alternate interior angles.

James Sousa: Angles and Transversals

James Sousa: Proof that Alternate Interior Angles Are Congruent

James Sousa: Proof of Alternate Interior Angles Converse

### Guidance

**Alternate Interior Angles** are two angles that are on the interior of

**Alternate Interior Angles Theorem:** If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

**Proof of Alternate Interior Angles Theorem:**

Given:

Prove:

Statement |
Reason |
---|---|

1. |
Given |

2. |
Corresponding Angles Postulate |

3. |
Vertical Angles Theorem |

4. |
Transitive PoC |

There are several ways we could have done this proof. For example, Step 2 could have been

**Converse of Alternate Interior Angles Theorem:** If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

#### Example A

Find

#### Example B

Find the measure of the angle and

The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for

#### Example C

Prove the Converse of the Alternate Interior Angles Theorem.

Given:

Prove:

Statement |
Reason |
---|---|

1. |
Given |

2. |
Vertical Angles Theorem |

3. |
Transitive PoC |

4. |
Converse of the Corresponding Angles Postulate |

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter3AlternateInteriorAnglesB

### Vocabulary

** Alternate Interior Angles** are two angles that are on the interior of

### Guided Practice

1. Is

2. What does

3. List the pairs of alternate interior angles:

**Answers:**

1. First, find

2. Because these are alternate interior angles, they must be equal for

3. Alternate Interior Angles:

### Practice

- Is the angle pair \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 3\end{align*} congruent, supplementary or neither?
- Give two examples of alternate interior angles in the diagram:

For 3-4, find the values of \begin{align*}x\end{align*}.

For question 5, use the picture below. Find the value of \begin{align*}x\end{align*}.

- \begin{align*}m\angle 4 = (5x - 33)^\circ, \ m\angle 5 = (2x + 60)^\circ\end{align*}

- Are lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} parallel? If yes, how do you know?

For 7-12, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

- \begin{align*}m\angle 4 = (3x-7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x-21)^\circ\end{align*}
- \begin{align*}m\angle 3 = (2x-1)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x-11)^\circ\end{align*}
- \begin{align*}m\angle 3 = (5x-2)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x)^\circ\end{align*}
- \begin{align*}m\angle 4 = (x-7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x-31)^\circ\end{align*}
- \begin{align*}m\angle 3 = (8x-12)^\circ\end{align*} and \begin{align*}m\angle 6 = (7x)^\circ\end{align*}
- \begin{align*}m\angle 4 = (4x-17)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x-29)^\circ\end{align*}

For questions 13-15, use the picture below.

- What is the alternate interior angle to \begin{align*}\angle 4\end{align*}?
- What is the alternate interior angle to \begin{align*}\angle 5\end{align*}?
- Are the two lines parallel? Explain.

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alternate exterior angles

Alternate exterior angles are two angles that are on the exterior of two different lines, but on the opposite sides of the transversal.alternate interior angles

Alternate interior angles are two angles that are on the interior of two different lines, but on the opposite sides of the transversal.### Image Attributions

Here you'll learn what alternate interior angles are and their relationship with parallel lines.