About Parallel Line and Transversal
- What are Parallel lines?
- What are Perpendicular lines?
- What are Transversals?
- What are Corresponding angles?
- What are Vertical angles?
- What are Supplementary angles?
- What are Interior angles?
- What are Exterior angles?
- What are Alternate interior angles?
- What are Alternate exterior angles?
- Solved Examples
- Frequently Asked Questions
What are Parallel lines?
Parallel lines are those lines that are always equidistant from each other and never meet no matter how far in either direction they are extended.
Parallel lines are represented by the symbol ||. WX || YZ, for example, denotes that line WX is parallel to line YZ.
What are Perpendicular lines?
Perpendicular lines are those that cross at a right angle (90 degrees). The symbol ⊥ is used to denote two perpendicular lines. WX ⊥ YZ, for example, denotes that line WX is perpendicular to line YZ.
What are Transversals?
A transversal is a line that crosses two lines in the same plane at two different points.
What are Corresponding angles?
Corresponding angles are formed when two parallel lines are intersected by a transversal. The pair of angles whose relative positions are the same at both intersections are called corresponding angles. Hence, corresponding angles are equal in measure if a transversal intersects 2 parallel lines.
In the figure below ∠1 and ∠2 form corresponding angles and hence ∠1 = ∠2.
What are Vertical angles?
When two lines intersect, vertical angles are formed as a pair. Because the angles are opposite each other, vertical angles are also known as vertically opposite angles. Not only that, vertical angles are equal in measure as well. We can use the concept of vertical angles to find the unknown angles in a figure. In the figure below, ∠a , ∠b and ∠c , ∠d form vertical angle pairs. Hence, ∠a = ∠b and ∠c = ∠d
What are Supplementary angles?
Angles that add up to 180 degrees are known as supplementary angles.
130° and 50°, for example, are supplementary angles because the sum of the two angles is 180°.
What are Interior angles?
Interior angles are angles that lie within a shape, or angles that lie within the area bounded by two parallel lines intersected by a transversal.
For example in the figure below, ∠a , ∠b and ∠c are interior angles of the triangle.
Also ∠1 , ∠2 , ∠3 and ∠4 are interior angles bounded by the two parallel lines
What are Exterior angles?
When a transversal cuts two parallel lines, 4 exterior angles are formed. These form two exterior angle pairs on the outside of the two parallel lines, with one pair on one side of the transversal and the second pair on the other side of the transversal.
Each pair of exterior angles formed by two parallel lines crossed by a transversal is supplementary.
For example in the figure below, ∠1, ∠7 and ∠2, ∠8 are the two pairs of exterior angles.
What are Alternate interior angles?
The alternate interior angles are those that are on the inner side of parallel lines but on opposite sides of the transversal. The alternate interior angles can be seen in the diagram below. Two parallel lines, AB and CD, are crossed by a transversal.
The pairs of alternate interior angles in the above figure, are:
- ∠4 and ∠6
- ∠3 and ∠5
The alternate interior angles are equal in measure, that is, ∠4 = ∠6 and ∠3 = ∠5.
What are Alternate exterior angles?
When a transversal intersects two or more parallel lines at different points, alternate exterior angles are formed. Formed on the outside of the two lines intersected by a transversal, alternate exterior angles are always on opposite sides of the transversal. Thus, the pair of alternate exterior angles is defined as the two exterior angles that form at the alternate ends of the transversals in the exterior part and are always equal. When a transversal cuts two parallel lines we get two such pairs of alternate exterior angles.
∠1, ∠7 and ∠ 2, ∠ 8 are alternate exterior angle pairs as both are located on the outside of the lines and on the opposite sides of the transversal.
Solved Parallel Line and Transversal Examples
Example 1: Find the angle measures of unknown angles in the figure?
Solution:
∠1 and the \(107^\circ \) angles are corresponding angles formed by a transversal intersecting parallel lines and the angles are congruent.
So, the measure of ∠1 is \(107^\circ \).
∠1 and ∠2 are supplementary angles. So, the sum of both angle will be \(180^\circ \).
∠1 + ∠2 = 180 Supplementary angles
\(107^\circ \) + ∠2 = 180 Substitute \(107^\circ \) for ∠1
∠2=\(73^\circ \) Subtract \(107^\circ \) from each side
So, the measure of ∠2 is \(73^\circ \).
Example 2: Find the angle measures of unknown numbered angles in the figure using corresponding angles?
Solution:
∠2 and the \(59^\circ \) angles are vertical angles. So, they are congruent.
So, the measure of ∠2 is \(59^\circ \).
∠1 and ∠2 are supplementary angles. So, the sum of both angle will be \(180^\circ \).
∠1 + ∠2 = 180 supplementary angles
∠1 + \(59^\circ \) + = 180 Substitute \(59^\circ \) for ∠1
∠1=\(121^\circ \) Subtract \(59^\circ \)from each side
So, the measure of ∠1 and ∠3 is \(121^\circ \).
∠1 and ∠4 are corresponding angles. So they are congruent.
Therefore, ∠4 = \(121^\circ \).
∠4 and ∠6 are vertical angles. So they are congruent.
Therefore, ∠6 = \(121^\circ \).
\(59^\circ \) and ∠5 are corresponding angles. So they are congruent.
Therefore,∠5 = \(59^\circ \).
∠5 and ∠7 are vertical angles. So they are congruent.
Therefore, ∠7 = \(59^\circ \).
Hence, the measures of ∠4 and ∠6 are \(121^\circ \), and the measures of ∠5 and ∠7 are \(59^\circ \).
Example 3: The image depicts a section of a highway. Explain the relationship between each angle pair.
(a) ∠1 and ∠4
∠1 and ∠4 are vertical angles. So, they are congruent.
(b) ∠2 and ∠4
∠2 and ∠4 are
supplementary angles. So, the sum of both angles will be \(180^\circ \).
(c) ∠3 and ∠6
The transversal intersecting parallel lines form alternate exterior angles ∠3 and ∠6.
(d) ∠2 and ∠7
The transversal intersecting parallel lines form alternate interior angles ∠2 and ∠7.
Example 4: The incline on the see-saw is 50 degrees. To keep a rail parallel to the incline of the steps, at what angles do you need to attach it to two parallel posts?
Solution:
Find the measures of ∠4, ∠5, ∠6 and ∠7 that make the rail parallel to the incline of the steps using angle relationships.
∠1: The \(50^\circ \) angle is complementary to ∠1.
∠1 + \(50^\circ \) = \(90^\circ \) complementary angles
∠1 = \(40^\circ \) Subtract \(50^\circ \)from each side
∠5: ∠1 and ∠5 are congruent because they are corresponding angles formed by a transversal intersecting parallel lines.
As a result, the value of ∠5 is 40.
∠4 and ∠5 are supplementary angles. So, the sum of both angle will be \(180^\circ \).
∠4 + ∠5 = 180 supplementary angles
∠4 + \(40^\circ \) = 180 Substitute \(40^\circ \) for ∠5
∠4 = \(140^\circ \) Subtract \(40^\circ \) from each side
So, the measure of ∠4 is \(140^\circ \).
∠6 and ∠7: Using alternate interior angles, the measure of ∠6 is \(40^\circ \) and the measure of ∠7 is \(140^\circ \).
You need to attach the rail so that the measure of ∠5 and ∠6 is \(40^\circ \) and the measure of ∠4 and ∠7 is \(140^\circ \).
Yes. A perpendicular transversal is a line that crosses parallel lines at right angles.
Yes. The two lines are parallel if a transversal intersects them in such a way that a pair of corresponding angles are equal.