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Alternate Interior Angles

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

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Alternate Interior Angles

Alternate Interior Angles 

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

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

Statement Reason
1. \begin{align*}l \ || \ m\end{align*} Given
2. \begin{align*}\angle 3 \cong \angle 7\end{align*} Corresponding Angles Postulate
3. \begin{align*}\angle 7 \cong \angle 6\end{align*} Vertical Angles Theorem
4. \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 \begin{align*}\angle 2 \cong \angle 6\end{align*} for the same reason, followed by \begin{align*}\angle 2 \cong \angle 3\end{align*}. We could have also proved that \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 \begin{align*}m \angle 1\end{align*}.

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

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

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

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

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

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

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

 

 

Examples 

Example 1

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

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

Example 2

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

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

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

Review

  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.

Review (Answers)

To view the Review answers, open this PDF file and look for section 3.4. 

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Vocabulary

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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