Corresponding Angles Theorem

Corresponding Angles: Definition, Examples & More 

Corresponding angles are formed in the matching or corresponding corners when a line (i.e. the transversal) intersects two parallel lines. Simply put, the corresponding angles occur when a third line cuts two parallel lines, and the angles are of equal measures at each intersection. So the first line angles are similar to those created by the second line with the transversal. Parallel railway tracks and solving a Rubik’s cube are two common examples of corresponding angles.

In the case of non-parallel lines, the corresponding angles formed by a transversal intersecting them have no relation. Even though all correspond to one another, they aren't equal. Similarly, there is no connection between the interior and exterior angles, vertically opposite and consecutive angles as far as the intersection of two non-parallel lines by a transversal is concerned.

Which Angles Are Corresponding Angles?

We have two parallel lines and a third line passing through them in this figure. The crossing line is transversal. Here, you’ll find eight angles forming four corresponding angle pairs in the diagram. For example:

• Angles 1 and 5 are one pair, while others are angles 3 and 7, angles 4 and 8, and finally, angles 2 and 6.
• Angles between the parallel lines like angles 2 and 8 are interior angles, whereas those outside the parallel lines i.e. 1 and 6, are exterior angles.
• Angles on the opposite sides like 1 and 8 are alternate angles.

Therefore, all angles – be it interior, exterior, alternate, or corresponding, are congruent.

For those wondering the meaning of 'congruent,' it means 'exactly equal' in shape and size. For instance, if you draw two circles with the same radius, cut them equally, and place them together, you'll find them perfectly together, meaning the circles are congruent.

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Types of Corresponding Angles

Corresponding angles are the angles formed on the opposite side of the transversal. So, please note that transversal intersects either two parallel or non-parallel lines. Therefore, it can be stated that corresponding angles are divided into two types:

• Corresponding angles formed by transversals and parallel lines: You will get corresponding angles with equal measures if a transversal cuts any two parallel lines. Hence, the angels created by the first line with transversal will get equal corresponding angles created by the second line with the transversal.
• Corresponding angles formed by transversals and non-parallel lines:In the case of non-parallel lines, the corresponding angles formed by a transversal intersecting them have no relation. Although they are not equal like in the parallel lines, they correspond to each other. Further, there's no connection between different angles like interior angles, exterior angles, consecutive angles, and vertically opposite angles. 

Corresponding Angles Theorem

The corresponding angles theorem suggests that when a line intersects two parallel lines, the corresponding angles in the two intersection regions are congruent. Thanks to this theorem, you already know that:

• All are right angles if one is a right angle.
• Four are acute angles; if one is acute
• Four are obtuse if one is an obtuse angle
• All angles can be termed adjacent, corresponding, and vertical angles. 

The Converse of Corresponding Angles Theorem

The converse of corresponding angles theorem states that if a transversal intersects two lines and their two intersection regions are congruent, they are parallel lines. Therefore, in case a transversal crosses the parallel lines and is similar to the corresponding angles set, the two intersected lines by a transversal are parallel. Thus, it is the converse of the corresponding angle theorem. 

Corresponding Angles Examples 

Let’s now take a look at some corresponding angles examples:

Example 1: Find the value of ‘x’ when the two corresponding angles are 5x + 10 and 55.

Answer: 

Considering that the given corresponding angles are congruent:

5x + 10 = 55

5x = 55 – 10

5x = 45

x = 9

Example 2: Find the magnitude of each corresponding angle when the values of two corresponding angles are 7y – 10 and 5y + 6.

Answer: 

Given values of the corresponding angles are 7y – 10 and 5y + 6, and they are congruent.

Hence, to find the value of ‘y’:

7y – 10 = 5y + 6

7y – 5y = 10 + 6

2y = 16

y = 8

The magnitude of each corresponding angle,

5y + 6 = 5 (8) + 6 = 46

7y – 10 = 7 * (8) – 10 = 46

Example 3: Find the value of ‘x’ when the values of two corresponding angles are ∠2 = 5x + 2 and ∠6 = 3x + 10.

Answer: 

As corresponding angles with parallel lines, the angles are congruent.

Equating the given expressions ∠2 = 5x + 1 and ∠6 = 2x + 10:

5x + 1 = 2x + 10

5x – 2x = 10 – 1

3x = 9

x = 9 / 3

x = 3

Therefore, the value of ‘x’ is 3.

And that's a wrap. Next, make sure you start Practicing your sums to get a clear idea of the fundamentals of the corresponding angles theorem. Good luck!

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