Corresponding Angles

Corresponding Angles are pairs of angles that occur at the same relative position at every intersection when a straight line crosses two other lines. For example, when two parallel lines are cut across by a transversal, it makes different pairs of angles "Equivalent" to one another .

1.0Corresponding Angles Types

Corresponding angles are separated into two categories based on whether the lines are parallel or not, which consist of: 

1. Corresponding Angles of Parallel Lines:

When two parallel lines are cut by a transversal, the corresponding angles of parallel lines are congruent; that is, they have equal measure. This is one of the most important properties of corresponding angles.

2. Corresponding Angles of Non-Parallel Lines:

When the lines cut by the transversal are not parallel, the corresponding angles are not equal. In such a case, the corresponding angles are formed, but they don't have the property of being congruent as they would with parallel lines.

2.0Theorems Related to Corresponding Angles

1. Corresponding Angles Theorem

The corresponding angles theorem postulate is used for defining an important property of two parallel lines crossed by a transversal. The theorem states that if a transversal intersects two parallel lines, then the corresponding angles formed on the same side of the transversal will be equal in measurement.

2. Converse of the Corresponding Angles

It is the reverse of the corresponding angles theorem, according to which if a transversal intersects two lines, making equal corresponding angles, then the two lines must be parallel. 

3.0Alternate and Corresponding Angles

Alternate and corresponding angles are some of the most important concepts of corresponding lines, with some important differences. Which includes: 

Alternate Angles	Corresponding Angles
Alternate angles lie on the opposite sides of a transversal.	Corresponding angles lie on the same side of the transversal but in the corresponding position.
Alternate angles are of two types, which are:  Alternate interior angels  Alternate exterior angles	The two types of corresponding angles are:  Corresponding Angles of Parallel Lines Corresponding Angles of Non-Parallel Lines

4.0Corresponding Sides and Corresponding Angles

The above section mentions the corresponding angles of two parallel lines crossed by a transversal. Now, in this section, we will discuss the Corresponding sides and angles in other shapes of geometry, including triangles. The terms corresponding sides and corresponding angles are quite commonly used while comparing shapes, especially triangles and other polygons. Let’s understand these in more detail:   

• Corresponding Angles in geometry: Corresponding angles in other geometric figures, such as triangles, refer to the angle that takes up the same relative position in two triangles. It is applied solely in congruence and similarity theorems. Example: If two triangles are congruent or similar, corresponding angles equal each other.

• Corresponding Sides: Another important concept of geometry is the corresponding sides of any two geometric figures. It refers to the sides of two geometric figures, for instance, triangles or quadrilaterals, in the same relative position. Therefore, corresponding sides are those which have the same length and also lie in the same position when compared to two congruent or similar figures.

5.0Corresponding Angles Examples

Problem 1: The two corresponding angles are given to be 6x + 20 and 80. What is the value of x?

Solution: Given that 6x+20 and 80 are two corresponding angles. Hence, 

6x + 20 = 80

6x = 80 – 20

x = 60/6

x = 10

Problem 2: The values of two corresponding angles are $\angle 9=3x+6and\angle 10=2x+15.$ Solve for the value of x.

Solution: As the given angles are corresponding angles hence, 

$\angle 9=\angle 10$

3x + 6 = 2x + 15

3x – 2x = 15 – 6

x = 9

Problem 3: The values of two corresponding angles are $\angle 4=5x+4and\angle 8=7x–6.$ Solve for the angles.

Solution: According to the question: 

$\angle 4=\angle 8$

5x + 4 = 7x – 6

5x – 7x = –6 – 4

–2x = –10

x = 5 

$\angle 4=5(5)–4=25–4=24$

$\angle 8=7(5)+6=35+6=42$