Alternate Exterior Angles

Chapter 37

Alternate Exterior Angles Concept Explanation:

Alternate exterior angles are the pairs of angles that lie on opposite sides of a transversal but outside the two lines it intersects. When the lines are parallel, these angles are equal.

For example, in the diagram below, $\angle 1$ and $\angle 7$ are alternate exterior angles, and so are $\angle 2$ and $\angle 8$ :

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\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] {$1$};
\draw (3.5,1.75) node[right] {$2$};
\draw (2.5,-0.25) node[left] {$7$};
\draw (3.5,-0.25) node[right] {$8$};
\end{tikzpicture}


*** Error message:
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Common Mistakes:

Confusing with Alternate Interior Angles: Students sometimes mistake alternate exterior angles for alternate interior angles.
Misidentifying the Transversal: Make sure the angles are on opposite sides of the transversal.

Helpful Tips:

Use Parallel Lines: When dealing with parallel lines, alternate exterior angles are always equal.
Label the Diagram: Clearly label the angles and the transversal to avoid confusion.

Hard Questions:

Q1: If $\angle 1 = 120^\circ$ , find $\angle 7$ .

Step-by-Step Solution:

Since $\angle 1$ and $\angle 7$ are alternate exterior angles, and the lines are parallel:

$\angle 7 = \angle 1 = 120^\circ$

Answer:

The measure of $\angle 7$ is $120^\circ$ .

Q2: In a diagram, if $\angle 2 = 85^\circ$ , what is $\angle 8$ ?

Step-by-Step Solution:

Since $\angle 2$ and $\angle 8$ are alternate exterior angles, and the lines are parallel:

$\angle 8 = \angle 2 = 85^\circ$

Answer:

The measure of $\angle 8$ is $85^\circ$ .

Q3: Two parallel lines are cut by a transversal. If $\angle 1 = 140^\circ$ , what is $\angle 7$ ?

Step-by-Step Solution:

Since $\angle 1$ and $\angle 7$ are alternate exterior angles, and the lines are parallel:

$\angle 7 = \angle 1 = 140^\circ$

Answer:

The measure of $\angle 7$ is $140^\circ$ .

Q4: If $\angle 7 = 110^\circ$ , what is $\angle 1$ ?

Step-by-Step Solution:

Since $\angle 1$ and $\angle 7$ are alternate exterior angles, and the lines are parallel:

$\angle 1 = \angle 7 = 110^\circ$

Answer:

The measure of $\angle 1$ is $110^\circ$ .

Q5: Given parallel lines and a transversal, if $\angle 8 = 75^\circ$ , what is $\angle 2$ ?

Step-by-Step Solution:

Since $\angle 2$ and $\angle 8$ are alternate exterior angles, and the lines are parallel:

$\angle 2 = \angle</li> </ol> 8 = 75^\circ$

Answer:

The measure of $\angle 2$ is $75^\circ$ .

Chapter 38: Alternate Exterior Angles Theorem

Alternate Exterior Angles Theorem Explanation:

The Alternate Exterior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. This theorem is a direct consequence of the properties of parallel lines and is crucial in solving problems involving transversal lines.

For example, if we have two parallel lines $l$ and $m$ intersected by a transversal $t$ , then the pairs of alternate exterior angles (like $\angle 1$ and $\angle 8$ , or $\angle 2$ and $\angle 7$ ) will always be equal.

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\begin{tikzpicture}
\draw<a href="0,0">thick</a> -- (6,0) node[right] {$l$};
\draw<a href="0,2">thick</a> -- (6,2) node[right] {$m$};
\draw<a href="3,-1">thick</a> -- (4,3) node[above] {$t$};
\draw (2.5,1.75) node[left] {$1$};
\draw (3.5,1.75) node[right] {$2$};
\draw (2.5,-0.25) node[left] {$8$};
\draw (3.5,-0.25) node[right] {$7$};
\end{tikzpicture}


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

Common Mistakes:

Confusing Interior with Exterior: Students often confuse alternate interior angles with alternate exterior angles.
Misidentifying Non-Parallel Lines: The theorem only applies when the lines are parallel. If they aren’t parallel, alternate exterior angles won’t be equal.

Helpful Tips:

Visualize the Diagram: Always draw or imagine the parallel lines and transversal to easily identify alternate exterior angles.
Check for Parallel Lines: Ensure that the lines are parallel before applying the theorem.

Hard Questions:

Q1: If $\angle 1 = 125^\circ$ , what is $\angle 8$ ?

Step-by-Step Solution:

By the Alternate Exterior Angles Theorem, $\angle 1 = \angle 8$ .

$\angle 8 = 125^\circ$

Answer:

The measure of $\angle 8$ is $125^\circ$ .

Q2: Given $\angle 2 = 95^\circ$ , find $\angle 7$ .

Step-by-Step Solution:

By the Alternate Exterior Angles Theorem, $\angle 2 = \angle 7$ .

$\angle 7 = 95^\circ$

Answer:

The measure of $\angle 7$ is $95^\circ$ .

Q3: Two parallel lines are cut by a transversal, and one alternate exterior angle is $135^\circ$ . What is the measure of the other alternate exterior angle?

Step-by-Step Solution:

By the Alternate Exterior Angles Theorem, the alternate exterior angles are congruent:

$\text{Alternate Exterior Angle} = 135^\circ$

Answer:

The other alternate exterior angle is also $135^\circ$ .

Q4: In a diagram, if $\angle 8 = 75^\circ$ , find $\angle 1$ .

Step-by-Step Solution:

By the Alternate Exterior Angles Theorem, $\angle 8 = \angle 1$ .

$\angle 1 = 75^\circ$

Answer:

The measure of $\angle 1$ is $75^\circ$ .

Q5: If $\angle 7 = 105^\circ$ , find the measure of $\angle 2$ .

Step-by-Step Solution:

By the Alternate Exterior Angles Theorem, $\angle 7 = \angle 2$ .

$\angle 2 = 105^\circ$

Answer:

The measure of $\angle 2$ is $105^\circ$ .

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