 # Corresponding Angles

### Introduction to Corresponding Angles

In plane geometry, Corresponding angles are formed when two lines are crossed by another line (which is known as Transversal). The angles created in matching corners at each intersection are the corresponding angles. In other words, a corresponding angle is one that holds on to the same correlative position simultaneously as another angle somewhere else in the figure. Corresponding angles are absolutely like one type of angle pair. As corresponding angles, you can have both alternate interior angles and alternate exterior angles.

In the figure given below;-

Example: P and Q are corresponding angles.

### Different types of Angles

1. Acute angle: An angle that measures any value between 0° and 90°

2. Obtuse angle: An angle that measures any value between 90° and 180°

3. Right angle: An angle that measures 90° is a Right angle

4. Straight angle: An angle that measures 180° is a straight angle

5. Supplementary angles: When the sum total of 2 angles is 180° then the angles are called supplementary angles. In addition, two right angles are always in supplementation to each other. The pair of adjacent angles whose sum is 180° is a linear pair.

6. Complementary angles: When the sum of 2 angles measures 90°, then these are called complementary angles.

7. Adjacent angles: The angles that have a common vertex and a common arm are called adjacent angles.

8. Vertically opposite angles: When two lines bisect, the angles that are created opposite to each other at the vertex (point of bisection) are called vertically opposite angles.

### Corresponding Angles and its Converse

Using Converse of the Corresponding Angles Postulate, you can prove lines are parallel. Therefore

We can take into account:

1. By Converse of the Corresponding Angles Postulate that implies that" If 2 lines and a transversal create corresponding angles that are in congruence, and then the two lines are parallel."

2. By Converse of the Alternate interior Angles Postulate that implies that "If 2 lines and a transversal create alternate interior angles that are in congruence, then the two lines are parallel."

3. By Converse of the Alternate Exterior Angles Theorem that implies that "If 2 lines and a transversal create an alternate exterior angles that are in congruence, then the two lines are parallel."

4. By Same-Side Interior Angles Principle that implies that "If 2 lines and a transversal create same-side interior angles that are additional (supplementary), then the two lines are parallel."

5. By Same-Side Exterior Angles and its Converse Principle that implies that "If 2 lines and a transversal create same-side exterior angles that are in congruence, then the two lines are parallel."

### Solved Examples

Problem 1:

Following is a plane figure with angle measures and naming in separate images? Can you tell Which Angles Are Corresponding Angles?

Solution 1:

Did you notice ∠ A corresponds to ∠ E? They make for a pair of corresponding angles. Now to find all the four pairs of corresponding angles in the figure, let’s use the corresponding angles theorem.

### Corresponding Angles Theorem

The Corresponding Angles Theorem says that: If a transversal line cuts the two parallel lines, eight angles are formed by three lines and their corresponding angles are congruent to each other. Thus, there are four pairs of corresponding angles which are as follows:-

1. ∠ A corresponds to ∠ E

2. ∠ B corresponds to ∠ F

3. ∠ C corresponds to ∠ G

4. ∠ D corresponds to ∠ H

Problem 2:

How to use corresponding angles to determine the values of different angles?

Example: In the diagram below, line ‘L’ is parallel to line ‘M’, and line “T’ is a transversal?

What is the measure of angle x?

1. 60 degrees

2. 80 degrees

3. 70 degrees

Solution 2:

By corresponding angles theorem, angles on the transversal line are corresponding angles which are equal. Thus exterior ∠ 110 degrees is equal to alternate exterior i.e. 110 degrees. X is adjacent. Making a semi-circle, the total area of angle measures 180 degrees. Thus

110 +x = 180

X = 180 – 110

X= 70 degrees.

Hence, answer is C. 70 degrees.

### Fun Facts

• Corresponding angles can never be adjacent angles.

• Corresponding angles do not touch each other, thus they can never be consecutive interior angles.

• Angles on the opposite side of the transversal are called alternate angles.

• Corresponding Angles are equal when the two lines are parallel.

• Two angles correspond to each other by being on the same side of the transversal. One is inside the parallel lines (an interior angle) and one is outside the parallel lines (an exterior angle).

Q.1 What is a Corresponding Angles Theorem?

Corresponding Angles Theorem comply with the following eight angles created by the three lines:

1. If one angle is a right angle, all are right angles.

2. If one angle is acute, other 4 are acute angles.

3. If one angle is obtuse, other four are obtuse angles.

4. All 8 angles can be categorized as adjacent angles, corresponding angles and vertical angles.

Q.2 What does Converse of the Corresponding Angle Postulate State?

Geometrically, the converse of the corresponding angle postulate describes that: If two lines and a transversal form relative or corresponding angles that are in congruence, then the two lines are parallel.

Q.3 What Happens when a Transversal Intersects two Parallel Lines?

• ü  The corresponding or relative angles are equal

• ü  The alternate interior and the alternate exterior angles are equal

• ü  The angles vertically opposite to each other are equal.

• ü  The pair of interior angles of the transversal that are on the same side is supplementary.

With that, we can conclude that the lines are parallel if we are able to verify at least one of the above mentioned conditions.

Q.4 How many Types of Angles are Formed by Transversal with two Lines?

1. Corresponding angles (4 pairs of relative angles).

2. Alternate interior angles ( 2 pairs of alternate interior angles).

3. Interior angles on the same side of transversal: (2 pairs of interior angles).

4. Alternate exterior angles (2 pairs of alternate exterior angles).