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Corresponding Angles

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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 l down to line m . 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 l || 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 then l || m .

Example A

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

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 B

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

\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 C

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

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

CK-12 Corresponding Angles

Guided Practice

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

2. Is l || m ?

3. Find the value of y :

Answers:

1. 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 104^\circ .

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

3. 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 .

Practice

  1. Determine if the angle pair \angle 4 and \angle 2 is congruent, supplementary or neither:
  2. Give two examples of corresponding angles in the diagram:
  3. Find the value of x :
  4. Are the lines parallel? Why or why not?
  5. Are the lines parallel? Justify your answer.

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

  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 .

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