Corresponding Angles: Definition, Theorems, Examples, FAQ

What Are Corresponding Angles?

According to the definition of corresponding angles, when a third line intersects two lines, the angles formed in matching corners or corresponding positions with the third line are known to be corresponding angles to each other.

In the above diagram:

Line 1 and Line 2 are two lines. Line 3 is intersecting Lines 1 and 2. 

The diagram shows that ∠1 and ∠2 have the same relative positions (upper right side angles). Therefore the angles ∠1 and ∠2 are corresponding angles.

Types of Corresponding Angles

We learnt that corresponding angles are formed on the opposite side of the transversal. The transversal can intersect either two parallel lines or two non-parallel lines. Hence, there can be two types of corresponding angles:

Corresponding Angles by Parallel Lines and Transversals

If a transversal (third line) intersects two parallel lines, then the corresponding angles formed are equal in the measurement. 

In the given diagram, two parallel lines are intersected by a transversal, forming 8 angles with the transversal. In this case, the angles formed by the first line with the transversal are congruent to corresponding angles formed by the second line with the transversal.

All corresponding angles in the diagram:

∠p, ∠w

∠q, ∠x

∠r, ∠y

∠s, ∠z

Since the corresponding angles formed by two parallel lines are congruent. So,

∠p = ∠w

∠q = ∠x

∠r = ∠y

∠s = ∠z

Corresponding Angles by Non-Parallel Lines and Transversals

If a transversal (third line) intersects any two non-parallel lines, then the corresponding angles formed are not equal in the measurement but are corresponding to each other.

Corresponding Angles Theorem

“If a third-line or a transversal intersects any two parallel lines, then the corresponding angles formed are equal in the measurement”.

The Corresponding Angles Converse Theorem

The converse will be the opposite to the above theorem and it says, “If the corresponding angles in the two intersection regions are equal in the measurement, then the two lines are parallel”.

Examples

1. The two corresponding angles formed by a transversal intersecting two parallel lines measure (9x + 6)° and 60°. Find the value of x?

In this case, the two corresponding angles will be congruent.

Hence,

(9x + 6)° = 60°

9x = (60 – 6)°

9x = 54°

x = (54/9)°

x = 6°

2. If ∠d = 30°, find the other angles in the diagram below?

Given, ∠d = 30°

∠d = ∠b (Vertically opposite angles)

So, ∠b = 30°

∠b = ∠ g = 30° (corresponding angles)

And, ∠d = ∠f (Corresponding angles)

So, ∠f = 30°

∠b + ∠a = 180° (supplementary angles)

∠a+ 30° = 180°

∠a = 180° – 30° = 150°

∠a = ∠c (Vertically opposite angles)

So, ∠c = 150°

∠ a = ∠ e (corresponding angles)

Hence, ∠e = 150°

 And, ∠c = ∠h = 150° (corresponding angles)

Therefore, ∠a = 150°, ∠b = 30° , ∠c = 150° , ∠e = 150° , ∠f = 30° , ∠g =30° and ∠h = 150°.