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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 using your knowledge of corresponding angles.

Watch This

Watch the portions of this video dealing with corresponding angles.

Watch this video beginning at the 4:50 mark.

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. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*} are corresponding angles.

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*} and both are cut by \begin{align*}t\end{align*}, then \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*}.

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

Investigation: Corresponding Angles Exploration

You will need: paper, ruler, protractor

1. Place your ruler on the paper. On either side of the ruler, draw lines, 3 inches long. This is the easiest way to ensure that the lines are parallel.
2. Remove the ruler and draw a transversal. Label the eight angles as shown.
3. Using your protractor, measure all of the angles. What do you notice?

In this investigation, you should see that \begin{align*}m \angle 1 = m \angle 4 = m \angle 5 = m \angle 8\end{align*} and \begin{align*}m \angle 2 = m \angle 3 = m \angle 6 = m \angle 7\end{align*}. \begin{align*}\angle 1 \cong \angle 4, \ \angle 5 \cong \angle 8\end{align*} by the Vertical Angles Theorem. By the Corresponding Angles Postulate, we can say \begin{align*}\angle 1 \cong \angle 5\end{align*} and therefore \begin{align*}\angle 1 \cong \angle 8\end{align*} by the Transitive Property.

Investigation: Creating Parallel Lines using Corresponding Angles
1. Draw two intersecting lines. Make sure they are not perpendicular. Label them \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, and the point of intersection, \begin{align*}A\end{align*}, as shown.
2. Create a point, \begin{align*}B\end{align*}, on line \begin{align*}m\end{align*}, above \begin{align*}A\end{align*}.
3. Copy the acute angle at \begin{align*}A\end{align*} (the angle to the right of \begin{align*}m\end{align*}) at point \begin{align*}B\end{align*}. See Investigation 2-2 in Chapter 2 for the directions on how to copy an angle.
4. Draw the line from the arc intersections to point \begin{align*}B\end{align*}.

From this construction, we can see that the lines are parallel.

Example A

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*}.

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 markings in the picture. By the Corresponding Angles Postulate the two angles are equal, so \begin{align*}m \angle 6 = 76^\circ\end{align*}.

Example C

Using the picture above, list pairs of corresponding angles.

Corresponding Angles: \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 7\end{align*}, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 5, \ \angle 4\end{align*} and \begin{align*}\angle 8\end{align*}

Watch this video for help with the Examples above.

Guided Practice

Lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} are parallel:

1. If \begin{align*}\angle 1=3x+1\end{align*} and \begin{align*}\angle 5 = 4x-3\end{align*}, solve for x.

2. If \begin{align*}\angle 2=5x+2\end{align*} and \begin{align*}\angle 6 = 3x+10\end{align*}, solve for x.

3. If \begin{align*}\angle 7=5x+6\end{align*} and \begin{align*}\angle 3 = 8x-10\end{align*}, solve for x.

1. Since they are corresponding angles and the lines are parallel, they must be congruent. Set the expressions equal to each other and solve for \begin{align*}x\end{align*}. \begin{align*}3x+1=4x-3\end{align*} so \begin{align*}x=4\end{align*}.

2. Since they are corresponding angles and the lines are parallel, they must be congruent. Set the expressions equal to each other and solve for \begin{align*}x\end{align*}. \begin{align*}5x+2=3x+10\end{align*} so \begin{align*}x=4\end{align*}.

3. Since they are corresponding angles and the lines are parallel, they must be congruent. Set the expressions equal to each other and solve for \begin{align*}x\end{align*}. \begin{align*}5x+5=8x-10\end{align*} so \begin{align*}x=5\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*}.

For questions 11-15, use the picture below.

1. What is the corresponding angle to \begin{align*}\angle 4\end{align*}?
2. What is the corresponding angle to \begin{align*}\angle 1\end{align*}?
3. What is the corresponding angle to \begin{align*}\angle 2\end{align*}?
4. What is the corresponding angle to \begin{align*}\angle 3\end{align*}?
5. Are the two lines parallel? Explain.

Vocabulary Language: English

Corresponding Angles

Corresponding Angles

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