Before jumping to corresponding angles, let us recall what are parallel lines, transversal lines, and non-parallel lines. Parallel lines are two or more lines that never intersect or meet in a two-dimensional plane. Whereas non-parallel lines intersect and meet at a point. A transversal line passes through the other two lines, regardless of their parallel nature.
Corresponding angles are the angles formed in corresponding corners with the transversal line. When two parallel lines intersect by any other line, i.e., the transversal, it creates corresponding angles. For example, angles p and w are the corresponding angles in the given figure. These are the angles that occupy a relative position at the intersection with the transversal. If these lines are parallel, the corresponding angles formed are also equal. These corresponding angles are a type of angle pair. These can have both alternate interior angles and alternate exterior angles.
Types of Angles:
Different types of Angles can form by the intersection of two or more lines. Let us discuss them briefly:
- Acute angle: An angle whose value lies between 0° and 90° an acute angle.
- Obtuse angle: An angle whose value lies between 90° and 180° is an obtuse angle.
- Right angle: An angle whose value is 90°, a right angle.
- Straight angle: An angle whose value is 180° is a straight angle.
- Supplementary angles: When the addition of two angles is equal to 180°, then the angles are called supplementary angles. Two right angles are always supplementary angles.
- Complementary angles: When the addition of two angles is equal to 90°, these angles are complementary angles.
- Adjacent angles: Adjacent angles are the angles that have a common vertex and a common arm.
- Vertically opposite angles: If two lines bisect, the angles created opposite to each other at the point of bisection are vertically opposite angles.
Types of Corresponding Angles
After understanding is corresponding angles, let us understand their types. We know that the transversal line can intersect two parallel or non-parallel lines. Thus, these angles are of two types:
1. Corresponding angles, including parallel lines and transversals
When a transversal line crosses two given parallel lines, the corresponding angles formed have equal measure. For example, the two parallel lines in the figure have a transversal intersecting them. It forms eight angles with the transversal line. So, the angles at the intersection of the first line with the transversal have equal corresponding angles formed by the intersection of the second line with the transversal. Hence,
- ∠p = ∠w
- ∠q = ∠x
- ∠r = ∠y
- ∠s = ∠z
∠p = ∠s, ∠q = ∠r, ∠w = ∠ z and ∠x = ∠y, are pairs of vertically opposite angles.
2. Corresponding angles, including non-parallel lines and transversals
When a transversal line intersects two non-parallel lines, the corresponding angles formed will not have any relation and will be unequal. They will be corresponding but not equal.
- Two corresponding angles cannot be adjacent angles.
- Two corresponding angles cannot be consecutive interior angles as they do not touch.
- The angles lying opposite to transversal are alternate angles.
- The two corresponding angles will be equal when the transversal line intersects two parallel.
- An interior and exterior angle correspond to each other by being on the same transversal side.
The types of Corresponding Angles according to the Sum?
They are of two types based on the sum. They are:
- The supplementary Corresponding Angles (when the sum is 180 degrees)
- The Complementary Corresponding angles (when the sum is 90 degrees)
Corresponding Angles Theory
This theory of the corresponding angle states that if the transversal line intersects two parallel lines, the corresponding angles are congruent. Moreover, the corresponding angles will always be equal if the transversal line crosses two lines that are parallel to each other.
Corresponding Angles in a Triangle
In a triangle, the angles of a congruent pair of sides of two congruent or identical triangles are corresponding angles. Therefore, these angles have the same value or are equal.
Corresponding Angle Proposition
This proposition or theorem of the corresponding angles states:
“When two parallel lines intersect a transverse line, then the angles in the regions of intersection are congruent and are corresponding angles.”
The Corresponding Angles Theorem Converse
The corresponding angle theorem works vice versa. So we can form the statement for the converse theorem as:
“If the intersection region angles are congruent and are corresponding angles, then the lines are parallel.” If a transversal intersects, the two lines are parallel. Then it forms the converse of the corresponding angle theorem.
Applications of Corresponding Angles
Corresponding angles have a wide range of applications that we often ignore. Let us study a few practical applications of corresponding angles.
- Usually, windows have grills in the form of square boxes or diamond blocks. They make corresponding angles.
- The bridge on the gigantic pillar stands strong because the pillars are connected in such a way that corresponding angles are equal.
- The railway tracks are professionally designed so that corresponding angles are equal.
Corresponding Angles Examples
Example 1: If the two corresponding angles are 6x + 12 and 70. Find the value of x?
Solution: Let the two angles be congruent corresponding angles.
6x + 12 = 70
6x = 70 – 12
6x = 58
x = 9.67
Example 2: Two corresponding angles are 8y – 15 and 6y + 7. What is the value of each corresponding angle?
Solution: Given values of corresponding angles are
8y – 15 and 6y + 7
We will now find the values of both the variables x and y.
We know that these are congruent corresponding angles.
8y – 15 = 6y + 7
8y – 6y = 15 + 7
2y = 22
y = 11
The magnitude of each corresponding angle,
8y – 15 = 8(11) – 15 = 73
6y + 7 = 6(11) + 7 = 73
Example 3: Given:
∠1 = 5x + 1 and ∠3 = 6x – 3, are two corresponding angles.
Find the value of x.
Solution: As these are corresponding angles, they will be congruent as lines are aid to be parallel.
We will now equate both the angles, and solve for x.
∠1 = 5x + 1 and ∠3 = 6x – 3,
5x + 1 = 6x – 3
1 + 3 = 6x – 5x
4 = x
Hence the value of x is 4.
Example 4: When two corresponding angles are ∠2 = 6x + 4 and ∠6 = 5x + 12. Find the value of x.
Solution: As these are corresponding angles so they will be congruent as lines are said to be parallel in nature.
We will now equate both the angles, and solve for x
∠2 = 6x + 4 and ∠6 = 5x + 12
6x + 4 = 5x + 12
6x – 5x = 12 – 4
x = 8
Hence the value of x is 8.
Example 5: When two corresponding angles are ∠7 = 5x + 6 and ∠3 = 9x – 10. Find the value of x.
Solution: As they are corresponding angles and the lines are parallel in nature, then they should be congruent.
Equate the given expressions ∠7 = 5x + 6 and ∠3 = 9x – 10 and find the value of x.
5x + 6 = 9x – 10
6 + 10 = 9x – 5x
16 = 4x
x = 16 / 4
x = 4
Hence the value of x is 4.