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# Corresponding Angles

## Angles in the same place with respect to a transversal, but on different lines.

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Corresponding Angles

What if you were presented with two angles that are in the same place with respect to the transversal but on different 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 and use corresponding angle postulates.

### Watch This

CK-12 Corresponding Angles

Watch the portions of this video dealing with corresponding angles.

James Sousa: Angles and Transversals

Then watch this video beginning at the 4:50 mark.

James Sousa: Corresponding Angles Postulate

Finally, watch this video.

James Sousa: Corresponding Angles Converse

### Guidance

Corresponding angles are two angles that are in the "same place" with respect to the transversal but on different lines. Imagine sliding the four angles formed with line \begin{align*}l\end{align*} down to line \begin{align*}m\end{align*}. The angles which match up are corresponding.

Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

If \begin{align*}l || m\end{align*}, then \begin{align*}\angle 1 \cong \angle 2\end{align*}.

Converse of Corresponding Angles Postulate: If corresponding angles are congruent when two lines are cut by a transversal, then the lines are parallel.

If then \begin{align*}l || m\end{align*}.

#### Example A

If \begin{align*}a || b\end{align*}, which pairs of angles are congruent by the Corresponding Angles Postulate?

There are 4 pairs of congruent corresponding angles:

\begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}, and \begin{align*}\angle 4 \cong \angle 8\end{align*}.

#### Example B

If \begin{align*}m\angle 2 = 76^\circ\end{align*}, what is \begin{align*}m\angle 6\end{align*}?

\begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} are corresponding angles and \begin{align*}l || m\end{align*} from the arrows in the figure. \begin{align*}\angle 2 \cong \angle 6\end{align*} by the Corresponding Angles Postulate, which means that \begin{align*}m\angle 6 = 76^\circ\end{align*}.

#### Example C

If \begin{align*}m\angle 8 = 110^\circ\end{align*} and \begin{align*}m\angle 4 = 110^\circ\end{align*}, then what do we know about lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*}?

\begin{align*}\angle 8\end{align*} and \begin{align*}\angle 4\end{align*} are corresponding angles. Since \begin{align*}m\angle 8 = m\angle 4\end{align*}, we can conclude that \begin{align*}l || m\end{align*}.

CK-12 Corresponding Angles

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### Guided Practice

1. Using the measures of \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} from Example B, find all the other angle measures.

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

3. Find the value of \begin{align*}y\end{align*}:

1. If \begin{align*}m\angle 2 = 76^\circ\end{align*}, then \begin{align*}m\angle 1 = 180^\circ - 76^\circ =104^\circ\end{align*} (linear pair). \begin{align*}\angle 3 \cong \angle 2\end{align*} (vertical angles), so \begin{align*}m\angle 3 = 76^\circ. \ m\angle 4 = 104^\circ\end{align*} (vertical angle with \begin{align*}\angle 1\end{align*}).

By the Corresponding Angles Postulate, we know \begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}, and \begin{align*}\angle 4 \cong \angle 8\end{align*}, so \begin{align*}m\angle 5 = 104^\circ, \ m\angle 6 = 76^\circ, \ m\angle 7 = 76^\circ\end{align*}, and \begin{align*}m\angle 104^\circ\end{align*}.

2. The two angles are corresponding and must be equal to say that \begin{align*}l || m\end{align*}. \begin{align*}116^\circ \neq 118^\circ\end{align*}, so \begin{align*}l\end{align*} is not parallel to \begin{align*}m\end{align*}.

3. The horizontal lines are marked parallel and the angle marked \begin{align*}2y\end{align*} is corresponding to the angle marked \begin{align*}80\end{align*} so these two angles are congruent. This means that \begin{align*}2y=80\end{align*} and therefore \begin{align*}y=40\end{align*}.

### Explore More

1. Determine if the angle pair \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 2\end{align*} is congruent, supplementary or neither:
2. Give two examples of corresponding angles in the diagram:
3. Find the value of \begin{align*}x\end{align*}:
4. Are the lines parallel? Why or why not?

For 6-10, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

1. If \begin{align*}m\angle 1 = (6x-5)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x+7)^\circ\end{align*}.
2. If \begin{align*}m\angle 2 = (3x-4)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x-10)^\circ\end{align*}.
3. If \begin{align*}m\angle 3 = (7x-5)^\circ\end{align*} and \begin{align*}m\angle 7 = (5x+11)^\circ\end{align*}.
4. If \begin{align*}m\angle 4 = (5x-5)^\circ\end{align*} and \begin{align*}m\angle 8 = (3x+15)^\circ\end{align*}.
5. If \begin{align*}m\angle 2 = (2x+4)^\circ\end{align*} and \begin{align*}m\angle 6 = (5x-2)^\circ\end{align*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.3.

### Vocabulary Language: English Spanish

Corresponding Angles

Corresponding Angles

Corresponding angles are two angles that are in the same position with respect to the transversal, but on different lines.