Lines and Angles

Parallel Lines and Transversal

Do you know what Parallel Lines are? You will understand this with the following examples. Every one of you must have seen the pair of railway tracks or a ladder or piano keys. What is one common thing among all these? The two tracks never meet each other, also the two sides of the ladder never intersect each other. The keys of the piano are always parallel to each other. Let us now study parallel and transversal lines and corresponding angles in detail.

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Parallel Lines

What are the lines and line segments?

A line is a straight path that is endless in both directions. That means it extends in both directions without end.  A line segment is a part of a line. The main difference the line and the line segment is that lines do not have endpoints while line segments have endpoints.

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What are Parallel Lines?

When the distance between a pair of lines is always the same, then we call such lines as parallel lines. The symbol for “parallel to” is “// “. Parallel lines are the lines which never meet each other. For the two lines to be parallel, the most important thing is that they are drawn in the same plane. These lines are always equidistant from each other.

What is a Transversal?

corresponding angles

A transversal is a line that passes through two lines lying in the same plane at two distinct points. In the transversal, the two given lines may be parallel or non-parallel. The angles formed when a transversal intersects two lines are as follows:

  • Corresponding Angles
  • Alternate Angles

1. Corresponding Angles

When two parallel or non-parallel lines in a plane are cut by a transversal, some angles are formed as shown in the figure below

corresponding angles

Here we have two lines that are parallel to each other. These two lines are the line a and line b. We can see one more line that intersects these two lines at two different points.  There are two points of intersection, which can see clearly in the above figure. As the transversal line intersects the two parallel line, we see that the angles below.

The following are the pairs of corresponding angles:

  • ∠ 1 and ∠ 6
  • ∠4 and ∠7
  • ∠2 and ∠5
  • ∠3 and ∠8

In all, we see that eight angles are formed here.   4 pairs of corresponding angles are formed and one important thing about these corresponding angles is that they are equal to each other, as the lines are parallel to each other.

2. Alternate Angles

When two parallel or non-parallel lines in a plane are cut by a transversal, some angles are formed as shown in the previous figure. The following are the pairs of alternate angles:

  • ∠ 4 and ∠5
  • ∠3 and ∠6

Properties of Transversal

A pair of parallel lines is intersected by a transversal. Following are the properties:

  • Vertically opposite angles are equal.
  • Corresponding angles are equal.
  • The interior angles formed on the same side of the transversal are supplementary.
  • Alternate angles are equal.

Solved Examples for You

Question 2: If l is any given line an P is any point not lying on l, then the number of parallel lines drawn through P, parallel to l would be:

corresponding angles

  1. One
  2. Two
  3. Infinite
  4. None of these

Answer : The correct option is A. Draw a line l and a point P not lying on l. Now we can draw a straight line parallel to l, which passes through P. We can see that only one line is drawn which is parallel to l and passes through P.

Question 2: Identify the given angle in the diagram.

corresponding angles

  1. Corresponding Angles
  2. Interior Angles
  3. Alterant Angles
  4. Alternate Exterior Angles

Answer : The correct option is D. The angles opposite to the sides of the transversal line and which is exterior is Alternate Exterior Angles.

Question 3: What is an example of a corresponding angle?

Answer: You already know that the transversal is when a line crosses two other lines, similarly, the angles in matching corners are referred to as corresponding angles. For instance, ‘a’ and ‘e’ are corresponding angles. Thus, when these two lines are parallel, the corresponding angles are equal.

Question 4: What is the sum of two corresponding angles?

Answer: As it is known that corresponding angles can be supplementary when the transversal intersects two parallel lines perpendicularly this is at 90 degrees. Thus, in such a case, each of the corresponding angles is going to be 90 degrees and their sum will add up to 180 degrees which is supplementary.

Question 5: What is a transversal?

Answer: A transversal refers to a line which passes through two lines lying in the same plane at two different points. Moreover, in the transversal, the two certain lines can be parallel or non-parallel. Thus, the angles which form when a transversal intersects two lines are corresponding angles and alternate angles.

Question 6: State the properties of a transversal.

Answer: The properties of a transversal are that first one being over here, the vertically opposite angles are equal. Further, the corresponding angles are equal and the interior angles which form on the same side of the transversal are supplementary. Finally, the alternate angles are equal.

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