About Alternate Angles
What are Alternate Angles?
A line that intersects two or more lines is called a transversal line.
When pairs of lines are intersected by a transversal, several pairs of angles are formed. We call the resultant angles as alternate angles, corresponding angles and vertical angles.
In figure (a), the line p and the line q are coplanar, but not parallel.
This means that p ∦ q, or line p is not parallel to line q, while line t remains the transversal. The various pairs of alternate angles are ∠3 and ∠6, ∠4 and ∠5, ∠1 and ∠8, ∠2 and ∠7.
When a pair of parallel lines are intersected by a transversal, pairs of congruent angles are formed, one of them being alternate angles. In figure (b), line p and line q are coplanar and parallel.
That means p ∥ q, indicating that line p is parallel to line q, and line t is the transversal.
The multiple pairs of alternate angles are ∠3 and ∠6 and ∠4 and ∠5, ∠1 and ∠8, ∠2 and ∠7 are congruent to each other.
Hence this indicates that, ∠3 = ∠6, ∠4 = ∠5, ∠1 = ∠8 and ∠2 = ∠7.
We can now conclude that when alternate angles are equal, the lines intersected by the transversal are always parallel.
Rapid Recall
∠3 and ∠6, ∠4 and ∠5 ∠1 and ∠8, ∠2 and ∠7
Solved Alternate Angles Examples
Example 1: The following image represents the intersection of highways. Identify the pairs of alternate angles.
Solution:
In the image, highway ‘a’ and highway ‘b’ are parallel to each other and intersected by the third highway.
Hence, the alternate angles shown in the image are:
∠3 = ∠6 and ∠1 = ∠8 [Alternate exterior angles]
∠4 = ∠5, and ∠2 = ∠7 [Alternate interior angles]
Example 2: A highway is intersected by a railway track as shown in the image, find the value of x.
Solution:
In the image, line ‘c’ and line ‘d’ form the rails of the railway track, which means, c ∥ d. The highway is the transversal.
From the figure , ∠1 is supplementary to ∠50°.
∠1 + 50° = 180° [Definition of supplementary angles]
∠1 = 180° – 50° [Subtract 50° each side]
∠1 = 130°
∠1 and ∠x are alternate interior angles, and so they are congruent.
∠1 = ∠x
130° = ∠x
The value of x is 130°.
Example 3: Find the value of x from the equations in the image.
Solution:
Since \(L_1\) and \(L_2\) are parallel, so (3x – 33)° and (2x + 26)° are alternate angles and hence, they are congruent.
(3x – 33)° = (2x + 26)° Write the equation
3x – 33 – 2x + 33 = 2x + 26 – 2x + 33 Subtract 2x and add 33 on each side
x = 59 Simplify
So the value of x is 59.
The angles interior to the parallel lines that lie on the same side of the transversal are called consecutive interior angles.
The two different types of alternate angles:
- Alternate interior angles
- Alternate exterior angles
The alternate interior angles are congruent. However, consecutive interior angles are supplementary.
Alternate interior angles are the angles located at the interior of the parallel lines and are on the opposite side of the transversal.