Transversal and Related Angles
Transversal is a straight line that crosses two or more other lines. The lines which the transversal crosses may or may not be parallel. When a transversal cuts two parallel lines, it makes 8 angles. When a transversal cuts n parallel lines, it makes 4 × n angles. Based on the positions of the angles above or below the transversal or to the left or right of transversal, the angle pairs are determined and classified into different categories like alternate interior angles, corresponding angles, alternate exterior angles, vertically opposite angles, etc.| 1. | What is a Transversal? |
| 2. | Classification of Angles on a Transversal |
| 3. | Transversals And Parallel Lines |
| 4. | Solved Examples |
| 5. | Practice Questions |
| 6. | FAQs on Transversals and Related Angles |
What is a Transversal?
For two or more lines, a transversal is any straight line that intersects two lines at distinct points. In the following figure, L1 and L2 are two lines that are cut at A and B by a transversal L0, resulting in a number of angles being formed. We observe that there are 8 angles formed.
Classification of Angles on a Transversal
There is a specific terminology associated with the angles formed when a transversal cuts two lines, as shown above. When two lines are intersected by a transversal, 8 angles are formed. Among these,
- The angles that lie in the area enclosed between two parallel lines are called interior angles. ∠3, ∠4 , ∠5, ∠6 are the interior angles.
- The angles that lie outside the area enclosed between two parallel lines are called exterior angles.∠1, ∠2 ,∠7 , ∠8 are the exterior angles.
These angles are classified into the following types based on their positions.
- Corresponding Angles
- Alternate Exterior Angles
- Alternate Interior Angles
- Co-Interior Angles
Corresponding Angles
When a transversal cuts two other lines, corresponding angles are the angles that occupy the same relative positions. Corresponding angles are on the same side of the transversal.
The following pairs of angles are the corresponding angles:
- ∠1 and ∠5 are both above the transversal and to the left of the transversal.
- ∠2 and ∠6 are both above the transversal and to the right of the transversal.
- ∠3 and ∠7 are both below the transversal and to the right of the transversal.
- ∠4 and ∠8 are both below the transversal and to the left of the transversal.
Alternate Interior Angles
Angles with different vertices, lying on the alternate sides of the transversal, and interior to the lines are called alternate-interior angles. Among the 4 interior angles, we find 2 pairs of angles that lie on the alternate sides of the transversals, having different vertices.The following pairs of angles are alternate interior angles:
- ∠3 and ∠5
- ∠4 and ∠6
Alternate Exterior Angles
Angles with different vertices, lying on the alternate sides of the transversal, and exterior to the lines are called alternate exterior angles. Among the 4 exterior angles, we find 2 pairs of angles that lie on the alternate sides of the transversals. The following pairs of alternate exterior angles are :
- ∠1 and ∠7
- ∠2 and ∠8
Co-interior Angles
Angles with different vertices, lying on the same sides of the transversal, are called co-interior angles. The following pairs of angles are co-interior angles or same-side interior angles:
- ∠3 and ∠6
- ∠4 and ∠5
Transversals and Parallel Lines
We will now go on to the specific case of two parallel lines being cut by a transversal. The following are the properties of the 8 angles so formed.
| All the corresponding angles are equal. |
∠1 = ∠5 ∠2 = ∠6 ∠4 = ∠8 ∠3 = ∠7 |
| All the alternate interior angles are equal. |
∠3 = ∠5 ∠4 = ∠6 |
| All the alternate exterior angles are equal. |
∠1 = ∠8 ∠2 = ∠7 |
| The co-interior angles are supplementary. |
∠3 +∠6 = 180° ∠4 + ∠5 = 180° |
Related topics on Transversals and Related Angles
Important Notes:
- A transversal is a line that cuts two or more lines at distinct points and the angles they make are related to each other based on their positions on the same side or the alternate sides of the transversal.
- The angles are classified as corresponding angles. alternate interior angles, alternate exterior angles and co-interior angles.
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Example 1: a) Which angle is alternate interior to ∠ 4?
b) If ∠5 = 110°, what is ∠3?
Solution:a) ∠4 is alternate interior to ∠6 because they are on the same side of the transversal and between two parallel lines.
b) If ∠5 = 110°, ∠3 = 180°- 110° = 70° because ∠5 and ∠3 are co-interior angles formed by the transversal while intersecting the two parallel lines. -
Example 2: What do you know about ∠1 and ∠2?
Solution:∠1 and ∠2 are on the alternate sides of the transversal and they are between the two lines that are cut by the transversal. Hence they are the alternate interior angles. But they are not equal as the lines on which they are drawn are not parallel to each other.
