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3.4: Corresponding Angles

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Identify corresponding angles as matching angles.

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 l down to line m. The angles which match up are corresponding.

f-d_e4af34d1b0947ec9e0c6e5d881d69d25e8e33ffddc2b11087f8241d5+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{1}

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

f-d_5cb183f69057304b506c6f83e8ea129defa8f490d1406072df9504df+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{2}

If l \parallel m, then \angle 1\cong \angle 2.

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

If

f-d_e07b6e10cf346793a9a35ddbbae1c31761ccab2ba7990215a74cb801+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{3}

then l \parallel m.

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?

Example \PageIndex{1}

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

f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{4}

Solution

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

Example \PageIndex{2}

Using the measures of \angle 2 and \angle 6 from Example 1, find all the other angle measures.

Solution

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

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

Example \PageIndex{3}

Find the value of y:

f-d_764c6382932c5ce076e8e9d394401386d1b5546376807aa3ce8ff988+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{5}

Solution

The horizontal lines are marked parallel and the angle marked 2y is corresponding to the angle marked 80 so these two angles are congruent. This means that 2y=80 and therefore y=40.

Example \PageIndex{4}

If a||b, which pairs of angles are congruent by the Corresponding Angles Postulate?

f-d_db29b4839d72e881911e626f7e7accc836a573e5d9a558caa2885019+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{6}

Solution

There are 4 pairs of congruent corresponding angles:

\angle 1\cong \angle 5, \angle 2\cong \angle 6, \angle 3\cong \angle 7, and \angle 4\cong \angle 8.

Example \PageIndex{5}

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

f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{7}

Solution

\angle 8 and \angle 4 are corresponding angles. Since m\angle 8=m\angle 4, we can conclude that l||m.

Review

  1. Determine if the angle pair \angle 4 and \angle 2 is congruent, supplementary or neither:
    f-d_c433bbb430c5a21eef1034baabdc9ea175e53f1b202b64c654c172d5+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{8}
  2. Give two examples of corresponding angles in the diagram:
    f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{9}
  3. Find the value of x:
    f-d_11ab2e84cc5141c83bc06315adff74666e742c4301bf7cdc8e8e87c6+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{10}
  4. Are the lines parallel? Why or why not?
    f-d_d704267123f6c08bfe461888971f9758ed5aab4aaa6477d2ed0c7221+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{11}
  5. Are the lines parallel? Justify your answer.
    f-d_19620e8aaf2620412ed42b0899b00aace1ba5472ca04b4405399e8d7+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{12}

For 6-10, what does the value of x have to be to make the lines parallel?

f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
Figure \PageIndex{13}
  1. If m\angle 1=(6x−5)^{\circ} and m\angle 5=(5x+7)^{\circ}.
  2. If m\angle 2=(3x−4)^{\circ} and m\angle 6=(4x−10)^{\circ}.
  3. If m\angle 3=(7x−5)^{\circ} and m\angle 7=(5x+11)^{\circ}.
  4. If m\angle 4=(5x−5)^{\circ} and m\angle 8=(3x+15)^{\circ}.
  5. If m\angle 2=(2x+4)^{\circ} and m\angle 6=(5x−2)^{\circ}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.3.

Resources

Vocabulary

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

Additional Resource

Interactive Element

Video: Corresponding Angles Principles - Basic

Activities: Corresponding Angles Discussion Questions

Study Aids: Angles and Transversals Study Guide

Practice: Corresponding Angles

Real World: Corresponding Angles


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