<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Alternate Interior Angles

## Angles on opposite sides of a transversal, but inside the lines it intersects.

Estimated9 minsto complete
%
Progress
Practice Alternate Interior Angles

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated9 minsto complete
%
Alternate Interior Angles

### Alternate Interior Angles

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.

#### Measuring Angles

1. 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.

2. 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*}

#### Proving the Converse

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

### Examples

#### Example 1

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

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.

#### Example 2

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

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*}

#### Example 3

List the pairs of alternate interior angles:

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*}.

### Review

1. Is the angle pair 6\begin{align*}\angle 6\end{align*} and 3\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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
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.