Introduction
Two lines on the same plane which never intersect and are equidistant from each other are called parallel lines. A line intersecting two or more lines is known as a transversal.
When a transversal line intersects two lines, corresponding angles are formed.
Corresponding angles can be formed in two ways:
- The intersection of two parallel lines and a transversal.
- The intersection of two non-parallel lines and a transversal.
Here we will discuss the corresponding angles formed by two parallel lines and a transversal in greater detail.
What are Corresponding Angles?
Corresponding angles are formed when a transversal intersects two parallel lines. These angles are located on the same side of the transversal in corresponding positions. Corresponding angles are always formed in pairs and are congruent.
In the figure, we can see that a transversal line is intersecting the parallel lines ‘a’ and ‘b’. So, as a result, pairs of corresponding angles are formed. There are a total of four pairs of corresponding angles.
Here, ∠1 and ∠5 are corresponding angles because both angles are on the corresponding corners on the left-hand side of the transversal.
Similarly, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8 are pairs of corresponding angles, so
∠1 = ∠5,
∠2 = ∠6,
∠3 = ∠7, and
∠4 = ∠8.
[Note: If corresponding angles formed by the intersection of two lines and a transversal are congruent, then the two lines are parallel.]
Learn About Angles Formed by Parallel Lines Cut by a Transversal in this Video
Rapid Recall
- Corresponding angles are congruent to each other.
- When the corresponding angles in the intersection region are congruent, the two lines are said to be parallel.
- If two parallel lines are cut by a transversal, each pair of corresponding angles is equal in measure.
- Corresponding angles lie on the same side of the transversal in corresponding positions.
Solved Corresponding Angles Examples
Example 1: The two corresponding angles are given to be (10x + 12)° and 52°. Find the value of x.
Solution:
Given,
The corresponding angles are (10x + 12)° and 52°.
We know that,
Corresponding angles are congruent to each other.
Therefore, 10x + 12 = 52
⇒ 10x = 52 – 12 [Transformation]
⇒ 10x = 40 [Subtract]
⇒ x = \(\frac{40}{10}\) [Divide]
⇒ x = 4
Hence, the value of x is 4.
Example 2: Find the corresponding angles in the figure:
Solution:
In the above figure, we can see that a transversal line ’GH’ is intersecting the parallel lines ‘AB’ and ‘CD’.
So, the corresponding angles of the given figure are as follows:
- ∠a and ∠e, because both angles are on the left-hand side of the transversal in the corresponding position.
- ∠b and ∠h, because both angles are on the right-hand side of the transversal in the corresponding position.
- ∠c and ∠f, because both angles are on the right-hand side of the transversal in the corresponding position.
- ∠d and ∠g are corresponding angles because both angles are on the left-hand side of the transversal.
Example 3: A pair of opposite angles measure (4x – 5)° and (8x – 10)°. Find the value of x.
Solution:
Given angles, (4x – 5)° and (8x – 10)°
We know that,
Corresponding angles are equal in measure.
Therefore, 4x – 5 = 8x – 10
⇒ 8x – 4x = 10 – 5 [Transformation]
⇒ 4x = 5 [Subtract]
⇒ x = \(\frac{5}{4}\) [Divide]
Hence, the value of x is \(\frac{5}{4}\).
Example 4: In the given figure, the stairs have a 45° incline. Find the measure of ∠1 and ∠5.
Solution:
It is given that the stairs have a 45° incline.
∠1 and ∠5 are corresponding angles.
As we know, corresponding angles are congruent, so,
∠1 = ∠5 …… (1)
To find the measure of ∠5, we have to find ∠1.
In the figure, ∠1 and 45° are complementary angles.
So, ∠1 + 4 = 90° [Definition of complementary angles]
1+45°=90° [Substitute]
∠1 = 45° [Subtract]
From equation (1) –
∠5 = 45°
Hence, ∠1 and ∠5 are corresponding angles, and their measure is 45°.
When a transversal intersects two parallel lines, alternate angles are formed on the opposite sides of the transversal. Alternate angles are congruent.
An angle whose measure is 90° is a right angle.
When a transversal intersects two non-parallel lines, the corresponding angles will not be equal. So, all corresponding angles are not equal.
Two lines intersecting at right angles, that is, 90°, are called perpendicular lines.