Formula to find the Measurement of different Angles in Parallel Lines
Parallel lines are lines that are in the same plane but they never intersect. A transversal is a line that intersects two or more lines. Hence, when a transversal runs across parallel lines, angles are formed. Note that when a transversal intersects two parallel lines, 8 angles are formed in total.
Consider two parallel lines n and m. Let a transversal t intersect these lines.
On intersection, ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8 are formed. ∠1, ∠2, ∠7 and ∠8 lie on the exterior of the parallel lines hence they are termed as exterior angles and ∠3, ∠4, ∠5 and ∠6 lie on the interior of the parallel lines hence are called interior angles.
Angles that are not next to each other and lie on opposite sides of the transversal are alternate angles. Alternate interior angles and alternate exterior angles are congruent.
Pairs of alternate interior angles: ∠3 = ∠6 and ∠4 = ∠5
Pairs of alternate exterior angles: ∠1 = ∠8 and ∠2 = ∠7
Corresponding angles that lie on the same side of the transversal are congruent. Hence, ∠1 = ∠5, ∠3 = ∠7, ∠2 = ∠6 and ∠4 = ∠8.
Solved Examples: Angles in Parallel Lines
Example 1: In the given figure, line m is parallel to line l. Also, line p is parallel to line q. Find x and y.
Solution:
In the given figure,
m ∥ l and q acts as the transversal.
∠EDC = ∠ABF (Corresponding angles)
2x + 60° = ∠ABF
∠ABF + ∠ABC = 180° (Supplementary angles)
x + 2x + 60° = 180°
3x = 180° – 60° = 120°
x = \(\frac{120^\circ}{3}\) = 40°
The points A, B and C form the vertices of a triangle. The sum of interior angles of a triangle is 180°.
∠ABC + ∠ACB + ∠BAC = 180°
40° + 40° + 5y = 180°
5y = 180° – 80° = 100°
y = \(\frac{100^\circ}{5}\) = 20°
Hence x is 40° and y is 20°
Example 2:
In the given figure, lines m ∥ p ∥ n. Find a.
Solution:
m ∥ p and AB acts as a transversal.
∠DAB = ∠ABE = 60° (Alternate interior angles)
∠ABC = 90° (Given in the figure)
So, ∠EBC = 90° – 60° = 30°
p ∥ n and BC acts as a transversal.
∠EBC = ∠BCG = 30° (Alternate interior angles)
Hence a is 30°.
Example 3:
In Le Parallelo town, every street has a consequent street that is parallel to it. The street view of a part of the town is given below.
The interior of the streets is filled with markets. The houses are situated at plots that are at the intersection of two streets. A house can be constructed if the respective angle in between intersecting streets is greater than 100°. For instance, in plot (D), the angle is 110° and this is greater than 100°. Hence a house can be constructed. Using all this information, find the value of x and also check if a house can be constructed at plot (C).
Solution:
7x = 2x + 50° (Vertical angles)
7x – 2x = 50°
5x = 50°
x = \(\frac{50^\circ}{5}\) = 10°
Therefore, 7x = \(7\times 10^\circ\) = 70°
The angle at point (A) and angle at point (C) form alternate exterior angles.
Therefore, the angle at point (C) is also 70°, which is lesser than 100°. Hence, a house cannot be constructed at point (C).
Example 4:
Check whether the lines m and n form parallel lines.
Solution:
r + 130° = 180° (Supplementary angles)
r = 180° – 130°
r = 50°
If lines m and n are parallel, then angle ‘r’ and the angle with a measurement of 60° form alternate interior angles.
Then r should be 60°, but r = 50°.
Therefore, lines m and n are not parallel.
When the sum of two angles add up to 180°, then the angles are said to be supplementary.
When the sum of two angles add up to 90°, then the angles are said to be complementary.
When two lines lie on the same plane but don’t intersect and are always at a constant distance from each other, they are said to be parallel.
Perpendicular lines are lines that intersect each other at right angles. That is, the angle between them would be 90°.
Interior angles on the same side of the transversal are co-interior angles. Co-interior angles add up to 180° and this is why co-interior angles are supplementary.