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# 3.4: Alternate Interior Angles

Difficulty Level: At Grade Created by: CK-12
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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

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

### Guidance

Alternate Interior Angles are two angles that are on the interior of l\begin{align*}l\end{align*} and m\begin{align*}m\end{align*}, but on opposite sides of the transversal. 3\begin{align*}\angle 3\end{align*} and 6\begin{align*}\angle 6\end{align*} are alternate interior angles.

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: l || m\begin{align*}l \ || \ m\end{align*}

Prove: 36\begin{align*}\angle 3 \cong \angle 6\end{align*}

Statement Reason
1. l || m\begin{align*}l \ || \ m\end{align*} Given
2. 37\begin{align*}\angle 3 \cong \angle 7\end{align*} Corresponding Angles Postulate
3. 76\begin{align*}\angle 7 \cong \angle 6\end{align*} Vertical Angles Theorem
4. 36\begin{align*}\angle 3 \cong \angle 6\end{align*} Transitive PoC

There are several ways we could have done this proof. For example, Step 2 could have been 26\begin{align*}\angle 2 \cong \angle 6\end{align*} for the same reason, followed by 23\begin{align*}\angle 2 \cong \angle 3\end{align*}. We could have also proved that 45\begin{align*}\angle 4 \cong \angle 5\end{align*}.

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 m1\begin{align*}m \angle 1\end{align*}.

m2=115\begin{align*}m \angle 2 = 115^\circ\end{align*} because they are corresponding angles and the lines are parallel. 1\begin{align*}\angle 1\end{align*} and 2\begin{align*}\angle 2\end{align*} are vertical angles, so m1=115\begin{align*}m \angle 1 = 115^\circ\end{align*} also.

1\begin{align*}\angle 1\end{align*} and the 115\begin{align*}115^\circ\end{align*} angle are alternate interior angles.

#### Example B

Find the measure of the angle and x\begin{align*}x\end{align*}.

The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for x\begin{align*}x\end{align*}.

(4x10)4xx=58=68=17\begin{align*}(4x-10)^\circ & =58^\circ\\ 4x & = 68^\circ\\ x & =17^\circ\end{align*}

#### Example C

Prove the Converse of the Alternate Interior Angles Theorem.

Given: l\begin{align*}l\end{align*} and m\begin{align*}m\end{align*} and transversal t\begin{align*}t\end{align*}

36\begin{align*}\angle 3 \cong \angle 6\end{align*}

Prove: l || m\begin{align*}l \ || \ m\end{align*}

Statement Reason
1. l\begin{align*}l\end{align*} and m\begin{align*}m\end{align*} and transversal t36\begin{align*}t \angle 3 \cong \angle 6\end{align*} Given
2. 32\begin{align*}\angle 3 \cong \angle 2\end{align*} Vertical Angles Theorem
3. 26\begin{align*}\angle 2 \cong \angle 6\end{align*} Transitive PoC
4. l || m\begin{align*}l \ || \ m\end{align*} Converse of the Corresponding Angles Postulate

Watch this video for help with the Examples above.

### Vocabulary

Alternate Interior Angles are two angles that are on the interior of l\begin{align*}l\end{align*} and m\begin{align*}m\end{align*}, but on opposite sides of the transversal.

### Guided Practice

1. Is l || m\begin{align*}l \ || \ m\end{align*}?

2. What does x\begin{align*}x\end{align*} have to be to make a || b\begin{align*}a \ || \ b\end{align*}?

3. List the pairs of alternate interior angles:

1. First, find m1\begin{align*}m \angle 1\end{align*}. We know its linear pair is 109\begin{align*}109^\circ\end{align*}. By the Linear Pair Postulate, these two angles add up to 180\begin{align*}180^\circ\end{align*}, so m1=180109=71\begin{align*}m \angle 1 = 180^\circ - 109^\circ = 71^\circ\end{align*}. This means that l || m\begin{align*}l \ || \ m\end{align*}, by the Converse of the Corresponding Angles Postulate.

2. Because these are alternate interior angles, they must be equal for a || b\begin{align*}a \ || \ b\end{align*}. Set the expressions equal to each other and solve.

3x+167035=5x54=2x=xTo make a || b, x=35.\begin{align*}3x+16^\circ&=5x-54^\circ\\ 70^\circ&=2x\\ 35^\circ&=x \qquad \quad \text{To make}\ a \ || \ b, \ x = 35^\circ.\end{align*}

3. Alternate Interior Angles: 4\begin{align*}\angle 4\end{align*} and 5\begin{align*}\angle 5\end{align*}, 3\begin{align*}\angle 3\end{align*} and 6\begin{align*}\angle 6\end{align*}.

### Practice

1. Is the angle pair \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 3\end{align*} congruent, supplementary or neither?
2. 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*}.

1. \begin{align*}m\angle 4 = (5x - 33)^\circ, \ m\angle 5 = (2x + 60)^\circ\end{align*}
1. 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?

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

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

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### Vocabulary Language: English

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.

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