Table of Contents
Last modified on August 3rd, 2023
Corresponding angles are pairs of angles that occupy the same relative position at each intersection when a transversal intersects two parallel straight lines.
The above figure shows two parallel lines AB and CD intersected by the transversal GH. The pairs of corresponding angles in the given figure are:
∠1 and ∠5
∠3 and ∠6
∠4 and ∠7
∠2 and ∠8
Corresponding angles are equal if the transversal intersects at least two parallel lines.
In the given figure, AB∥CD,
Thus,
∠1 = ∠5, ∠3 = ∠6, ∠4 = ∠7, ∠2 = ∠8
Corresponding angles formed when a transversal intersects at least two non-parallel lines are not equal and are also found to have no relation with each other.
The two types of corresponding angles are:
1) Corresponding Interior Angle: Found at the inner side of the intersection between the parallel lines and the transversal.
2) Corresponding Exterior Angle: Found at the outer side of the intersection between the parallel lines and the transversal.
Prove Corresponding Angles Theorem
∠1 = ∠5, ∠3 = ∠6, ∠4 = ∠7, ∠2 = ∠8
Given, PQ and RS are the two parallel lines intersected by the transversal IJ.
Now, if PQ∥ RS
Then by corresponding angles theorem, we can write
∠1 = ∠5
∠3 = ∠6
∠4 = ∠7
∠2 = ∠8
Hence Proved
Remember: Thus, the only way to prove corresponding angles congruent is when the given lines are parallel. Thus, the Corresponding Angles Theorem is true without proof.
Prove the Converse of Corresponding Angles Theorem
PQ∥ RS
Given, ‘PQ’, ‘RS’ are two lines intersected by the transversal ‘IJ’, and ∠1 = ∠5, ∠3 = ∠6, ∠4 = ∠7, ∠2 = ∠8 are the pairs of corresponding angles
Then, by the Converse of Interior Angles Theorem,
PQ∥ RS
Hence Proved
Remember: Thus, the only way to prove the given lines parallel is just by stating the corresponding angles congruent. Thus the Converse of Corresponding Angles Theorem is accepted as true without proof.
Let us take some examples to understand the concept better.
Find the missing angles in the given diagram. Given that ∠3 = 45°
Given, ∠3 = 45°
∠3 = ∠4 (Vertically opposite angles)
Hence, ∠4 = 45°
Now,
∠4 =∠7 = 45° (Corresponding Angles)
∠3 =∠6 (Corresponding Angles)
Hence, ∠6 = 45°
Now,
∠1 + ∠4 = 180° (Supplementary angles)
∠1 + 45° = 180°
∠1 = 180° – 45°
∠1 =135°
Similarly,
∠1 = ∠2 (Vertically opposite angles)
Hence, ∠2 = 135°
Again,
∠1 = ∠5 (Corresponding Angles)
Hence, ∠5 = 135°
Similarly,
∠2 = ∠8 (Corresponding Angles)
Hence, ∠8 = 135°
The two corresponding angles measure (2x +10)° and 70°. Find the value of x.
According to the corresponding angles theorem, the two corresponding angles are congruent.
Thus,
(2x +10)° = 70°
2x = 70° – 10°
x = 60°/2
x = 30°
The two corresponding angles of the given figure is 6y-14 and 4y + 6. Find the magnitude of a corresponding angle.
According to the corresponding angles theorem, the two corresponding angles are congruent.
Thus,
6y-14 = 4y + 6
6y – 4y = 6 + 14
2y = 20
y = 10
Thus, the magnitude is,
6y-14 = 6 x 10 – 14 = 46°
Ans. Corresponding angles are supplementary only if the transversal perpendicularly intersects two parallel lines.
Last modified on August 3rd, 2023
Great explanation of corresponding angles! I learned a lot from this post. The examples made it easy to understand and visualize the concept. Thank you for sharing!