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Lines and Angles

Related Angles

Related angles are the pairs of angles and specific names are given to the pairs of angles which we come across. These are called related angles as they are related to some condition. In this section, we’ll learn all about related angles and so on.

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Angles Made by a Transversal

When any two lines are cut by a transversal, then eight angles are formed as shown in the adjoining figure. The angles so formed have been given specific names. The following table gives the types of angles and their names in reference to the adjoining figure.

  1. Interior angles

    1. Whose one of the arms includes the transversal,
    2. Lie inside the region between the two straight lines.
  2. Exterior angles

    1. Whose one of the arms includes the transversal,
    2. Lie outside the region between the two straight lines.
  3. Co-interior angles

    1. are the interior angles lying on the same side of the transversal.
    2. have different vertices.
  4. Corresponding Angles

    1. have different vertices,
    2. are on the same side of the transversal and
    3. are in the corresponding position like above or below, left or right relative to the two lines.
  5. Alternate interior angles

    1. have different vertices,
    2. are on opposite sides of the transversal and
    3. lie between the two lines.
  6. Alternate exterior angles

    1. have different vertices,
    2. Are on opposite sides of the transversal and
    3. do not lie between the two lines.

Here is an example –

Related Angles

  • Interior angles = angle 2, angle 3, angle 5, angle 8
  • Exterior angles = angle 1, angle 4, angle 6, angle 7
  • Corresponding angles = angle 1 and angle 5, angle 2 and angle 6, angle 4 and angle 8, angle 3and angle 7
  • Alternate interior angles = angle 3 and angle 5, angle 2 and angle 8
  • Alternate exterior angles = angle 1 and angle 7, angle 4 and angle 6
  • Consecutive interior angles or co-interior angles = angle 2 and angle 5, angle 3 and angle 8

Angles Made by a Transversal to Two Parallel Lines

If two parallel lines are cut by a transversal, then

  1. Each pair of corresponding angles is equal in measure.
  2. Each pair of alternate interior angles is equal in measure.
  3. Also, each pair of interior angles on the same side of the transversal are supplementary, i.e., co-interior angles are supplementary.

F and Z Shapes

To locate corresponding angles when the parallel lines are intersected by a transversal, look for the shape of F. As you can see in the figures that follow, the shape of F is formed in four different positions. As a result, four corresponding pairs of angles are formed.

Related Angles

In the same way, when two parallel lines are intersected by a transversal, the shape of Z formed shows alternate interior angles.

Related Angles

Note:

  • The F-shape shows corresponding angles.
  • The Z-shape shows alternate interior angles.

Checking for parallel lines. You know that, when two lines are parallel, then a transversal gives rise to equal corresponding angles, equal alternate interior angles, and co-interior angles being supplementary. Conversely, If any two lines are cut by a transversal such that

  1. Any pair of corresponding angles are equal or
  2. Any pair of alternate interior angles are equal or
  3. Co-interior angles are supplementary, then the two lines are parallel.

Solved Example for You on Related Angles

Question: Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the larger angle is:

  1. 72°
  2. 108°
  3. 36°
  4. 54 °

Solution: D. Let the complementary angles be x and 90 ° –  x

Then, 2= 3 (90 ° –  x)

2= 270 º – 3⇒ 5x = 270 º

=  54 °

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