In geometry, a transversal is a line that intersects two or more lines at distinct points in the same plane. The lines that are intersected by the transversal may be parallel or non-parallel.
When a transversal intersects two parallel lines, several pairs of special angles are formed. In the given figure, lines 1 and 2 are parallel lines, and the slanted line intersecting them is the transversal.
The intersection of the transversal with the two parallel lines forms eight angles: a, b, c, d, q, p, r, and s. These angles create different angle pairs such as:
- Corresponding angles (a & p, b & q, c & r, d & s)
- Alternate interior angles (c & p, d & q)
- Alternate exterior angles (a & r, b & s)
- Vertically opposite angles (a & c, b & d, q & s, p & r)
Understanding these angle relationships helps in solving many geometry problems involving parallel lines and transversals.
Corresponding Angles
Corresponding angles are the angles that lie in the same relative position at each intersection of the transversal with the two lines. If the two lines are parallel, corresponding angles are equal. The following pairs of angles from the figure above are corresponding angles:
- ∠a = ∠p
- ∠b = ∠q
- ∠d = ∠s
- ∠c = ∠r
Alternate Exterior Angles
Alternate exterior angles are the angles that lie outside the two parallel lines and on opposite sides of the transversal. When the lines are parallel, these angles are equal.
The following pairs of angles from the figure above are alternate exterior angles:
- ∠a = ∠r
- ∠b = ∠s
Alternate Interior Angles
Alternate interior angles are the angles that lie between the two parallel lines and on opposite sides of the transversal. If the lines are parallel, these angles are equal.
The following pairs of angles from the figure above are alternate interior angles:
- ∠d = ∠q
- ∠c = ∠p
Vertically Opposite Angles
Vertically opposite angles are the angles that are formed opposite each other when two lines intersect. These angles are always equal. The following pairs of angles from the figure above are vertically opposite angles:
- ∠a = ∠c
- ∠b = ∠d
- ∠p = ∠r
- ∠q = ∠s
Constructing a Transversal on Parallel Lines
Constructing a transversal line on a parallel line is very simple. To construct a transversal line between two parallel lines follow the the steps added below,
Step 1: Take a pair of Parallel Lines (say 1 and 2)
Step 2: Draw a line (t) that cuts the first line.
Step 3: Now extend the line and then make it cut the second line (2) then the line so obtained is the transversal line.
Solved Examples
Example 1: In Figure, if PQ || RS, ∠ MXQ = 135° and ∠ MYR = 40°, find ∠ XMY.
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.
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.
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.
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°