Alternate Interior Angles

Chapter 39: Alternate Interior Angles

Concept Explanation:

Alternate Interior Angles are pairs of angles that lie on opposite sides of a transversal but inside the two lines it intersects. When the lines are parallel, alternate interior angles are congruent.

For example, in the diagram below, $\angle 3$ and $\angle 6$ , and $\angle 4$ and $\angle 5$ are pairs of alternate interior angles:

*** QuickLaTeX cannot compile formula:

\begin{tikzpicture}
\draw<a href="0,0">thick</a> -- (6,0);
\draw<a href="0,2">thick</a> -- (6,2);
\draw<a href="3,-1">thick</a> -- (4,3);
\draw (2.5,1.75) node[left] {$3$};
\draw (3.5,1.75) node[right] {$4$};
\draw (2.5,-0.25) node[left] {$6$};
\draw (3.5,-0.25) node[right] {$5$};
\end{tikzpicture}


*** Error message:
Error: Cannot create svg file

Common Mistakes:

Confusing Interior with Exterior: Students sometimes mix up alternate interior and exterior angles.
Using Non-Parallel Lines: The property only holds when the lines are parallel.

Helpful Tips:

Label Your Diagram: Clearly label the angles when working with transversals to avoid confusion.
Use a Parallel Line Check: Always confirm the lines are parallel.

Hard Questions:

Q1: If $\angle 3 = 120^\circ$ , what is $\angle 6$ ?

Step-by-Step Solution:

Since $\angle 3$ and $\angle 6$ are alternate interior angles, and the lines are parallel:

$\angle 6 = \angle 3 = 120^\circ$

Answer:

The measure of $\angle 6$ is $120^\circ$ .

Q2: Given $\angle 4 = 85^\circ$ , find $\angle 5$ .

Step-by-Step Solution:

Since $\angle 4$ and $\angle 5$ are alternate interior angles, and the lines are parallel:

$\angle 5 = \angle 4 = 85^\circ$

Answer:

The measure of $\angle 5$ is $85^\circ$ .

Q3: Two parallel lines are cut by a transversal. If $\angle 6 = 75^\circ$ , find $\angle 3$ .

Step-by-Step Solution:

Since $\angle 3$ and $\angle 6$ are alternate interior angles, and the lines are parallel:

$\angle 3 = \angle 6 = 75^\circ$

Answer:

The measure of $\angle 3$ is $75^\circ$ .

Q4: If $\angle 5 = 95^\circ$ , what is $\angle 4$ ?

Step-by-Step Solution:

Since $\angle 4$ and $\angle 5$ are alternate interior angles, and the lines are parallel:

$\angle 4 = \angle 5 = 95^\circ$

Answer:

The measure of $\angle 4$ is $95^\circ$ .

Q5: In a parallel line diagram, if $\angle 6 = 105^\circ$ , find $\angle 3$ .

Step-by-Step Solution:

Since $\angle 3$ and $\angle 6$ are alternate interior angles, and the lines are parallel:

$\angle 3 = \angle 6 = 105^\circ$

Answer:

The measure of $\angle 3$ is $105^\circ$ .

Chapter 40: Alternate Interior Angles Theorem

Concept Explanation:

The Alternate Interior Angles Theorem states that when two parallel lines are cut by a transversal, each pair of alternate interior angles is congruent. This theorem is widely used to prove that lines are parallel or to find unknown angle measures.

For example, if lines $l$ and $m$ are parallel, and transversal $t$ cuts across them, then $\angle 3$ is congruent to $\angle 6$ , and $\angle 4$ is congruent to $\angle 5$ .

*** QuickLaTeX cannot compile formula:

\begin{tikzpicture}
\draw<a href="0,0">thick</a> -- (6,0);
\draw<a href="0,2">thick</a> -- (6,2);
\draw<a href="3,-1">thick</a> -- (4,3);
\draw (2.5,1.75) node[left] {$3$};
\draw (3.5,1.75) node[right] {$4$};
\draw (2.5,-0.25) node[left] {$6$};
\draw (3.5,-0.25) node[right] {$5$};
\end{tikzpicture}



*** Error message:
Error: Cannot create svg file

Common Mistakes:

Misidentifying Angles: Students sometimes mistake alternate interior angles for corresponding or exterior angles.
Applying the Theorem to Non-Parallel Lines: This theorem is only valid when the lines are parallel.

Helpful Tips:

Understand Angle Pairs: Familiarize yourself with different angle pairs (e.g., corresponding, alternate exterior, and interior).
Ensure Parallelism: Always check if the lines are parallel before applying the theorem.