Transversal Line Examples
Example 1: In Figure, if PQ || RS, ∠ MXQ = 135° and ∠ MYR = 40°, find ∠ XMY.
Solution:
Let’s construct a line AB parallel to line PQ, through point M.
Now, AB || PQ and PQ || RS
AB || RS || PQ
∠ QXM + ∠ XMB = 180°…(Interior Angles on the Same Side of Transversal are Supplementary)
∠ QXM = 135°…(given)
135° + ∠ XMB = 180°
∠ XMB = 45°
∠ BMY = ∠ MYR…(Alternate Angles)
∠ BMY = 40°
∠ XMB + ∠ BMY = 45° + 40°
Therefore, ∠ XMY = 85°
Example 2: In Figure, AB || CD and CD || EF. Also EA ⊥ AB. If ∠ BEF = 55°, find the values of x, y, and z.
Solution:
Since,
AB || CD and CD || EF
AB || CD || EF
EB and AE are Transversal
y + 55° = 180°…(Interior Angles on the Same Side of Transversal are Supplementary)
y = 180° – 55° = 125°
x = y (Corresponding Angles)
x = y = 125°
Now, ∠ EAB + ∠ FEA = 180°…(Interior Angles on the Same Side of Transversal are Supplementary)
90° + z + 55° = 180°
Hence, z = 35°
Example 3: In Figure, find the values of x and y and then show that AB || CD.
Solution:
Here,
x + 50° = 180° (Linear Pair is equal to 180°)
x = 130°
y = 130° (Vertically Opposite Angles are Equal)
x = y = 130°
In two parallel lines, the alternate interior angles are equal, and ∠x = ∠y
Hence, this proves that alternate interior angles are equal and so, AB || CD
Example 4: In Figure, if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y.
Solution:
Here, ∠APQ = ∠PQR…(Alternate Interior Angles)
x = 50°
∠APR = ∠PRD…(Alternate Interior Angles)
∠APQ + ∠QPR = 127°
127° = 50°+ y
y = 77°
Hence, x = 50° and y = 77°
Transversal Lines
Transversal Lines in geometry is defined as a line that intersects two lines at distinct points in a plane. The transversal line intersecting a pair of parallel lines is responsible for the formation of various types of angles that, include alternate interior angles, corresponding angles, and others. A transversal can intersect at least two lines that can be parallel or non-parallel.
Here, in this article, we learn about Parallel Lines and Transversals, Angle Relationship Between Parallel Lines and Transversals, and others in detail.
Table of Content
- What is a Transversal Line?
- Transversal Lines and Parallel Lines
- Transversal Lines and Angles
- Constructing a Transversal on Parallel Lines
- Transversal Line Examples