Corresponding Angles: Theorem & Examples

Angle is formed by two rays that have a common end point. There are some types of angle. Here they are.

Type of Angle

In general, there are some kinds of angles:

1. Acute angle (less than 90°)

Acute Angle

Learn more about acute angle.

2. Right angle (90°)

Right Angle

3. Obtuse angle (more than 90° but less than 180°)

Obtuse Angle

Learn more about obtuse angle.

4. Straight angle (180°)

Straight Angle

5. Reflect angle (more than 180°)

Reflect Angle

Learn more about reflect angle.

If two parallel lines are cut by another line (transversal) then there are angles that have same measure (same magnitude) which is called corresponding angles.

Corresponding Angels Theorem

Corresponding angles theorem says that

“If a transversal cuts two parallel lines, their corresponding angles are congruent”

Based on the theorem, there is eight angles that can be used to solve the angle’s problems

For more details, look at the picture.

Corresponding Angles Theorm

Based on the picture, lines a parallel with line b and they are cut by line c so there are angles that have same measure (same magnitude) which is called corresponding angles.

Based on the theorem, there are:

1. Alternate interior angles

Pair of alternate interior angles has same measure. Based on the picture, alternate interior angles are ∠a4 = ∠b2, and ∠a4 = ∠b2.

2. Alternate exterior angles

Pair of alternate exterior angles has same measure. Based on the picture, alternate interior angles are ∠a1 = ∠b3, and ∠a2 = ∠b4.

3. Consecutive interior angles

Sum of the pair of consecutive interior angles is 180°. They are ∠a4 + ∠b1 = 180° and ∠a3 + ∠b2 = 180°.

4. Consecutive exterior angles

Sum of the pair of consecutive exterior angles is 180°. They are ∠a1 + ∠b4 = 180° and ∠a2 + ∠b3 = 180°.

5. Corresponding angles

Pair of corresponding angles has same measure. Based on the picture, they are ∠a1 = ∠b1, ∠a2 = ∠b2, ∠a3 = ∠b3, and ∠a4 = ∠b4.

6. Vertical angles

Pair of vertical angles has same measure. Based on the picture, they are ∠a1 = ∠a3, ∠a2 = ∠a4, ∠b1 = ∠b3, and ∠b2 = ∠b4. It is also called ∠a1 is opposite of ∠a3.

Corresponding Angles Examples

  1. Find the a, b, and c!
Corresponding Angles Examples
Answer

Using corresponding angles theorem,

  • ∠x has same measure with angles in the opposite, so it is x (vertical angles).
  • ∠x (based on the previously reason) and ∠(x+40)° are consecutive interior angles so the sum is 180°.

x + 40o + x = 180o

2x + 40o = 180o

2x = 140o

x = 70o

  • ∠a is opposite of ∠(x+40)° so it has same measure, ∠a = 70o + 40o = 110o
  • The sum of ∠b and ∠a is 180°. It is because they are supplementary angles.

Besides that, ∠b can be found by using reason that ∠b is supplementary with ∠(x+40)° so the sum is 180°, or using reason that ∠b and ∠x are alternate interior angles so they have same measure.

2. Find the x!

Corresponding Angles Examples 2
Answer

Making another line between the two parallel lines in the picture make it solve easily. Called it line a, b, and c.

Corresponding Angles Examples 3
  • Because of ∠y is supplementary angle with ∠30° so ∠y = 150° (we assume that there is ∠y)
  • Focus on lines b and c, so we have :

(Using alternate interior angles and consecutive interior angles)

Corresponding Angles Examples 4
  • Focus on lines a and b (Using consecutive interior angles)
Corresponding Angles Examples 5
  • Now, look on the x. it is sum of angles on the top and bottom of line b.

x = 45o + 30o = 75o

Learn More

Exterior Angle Theorem

Consecutive Interior Angles

Supplementary Angles