Corresponding Angle and Alternate Angle Theorems

Corresponding and Alternate angles are the most fundamental angles formed when two parallel lines are intersected by a transversal line. In this article, we will discuss the Corresponding and Alternate Angle Theorems in detail along with their proof in detail. Corresponding and Alternate Angles are the most widely used tool in Geometry. Parallel lines find many applications in our day-to-day life and so do the properties of parallel lines such as corresponding and alternate angles. Corresponding and Alternate Angles theorems form a fundamental tool of Euclidean Geometry.

Statement of Corresponding and Alternate Angle Theorems

Alternate Angle Theorem

The theorem states that if a line referred to as transversal intersects parallel lines, the alternate interior angles are congruent.

Corresponding Angle Theorem

The theorem states that if a transversal intersects parallel lines, the corresponding angles are congruent.

Angles made by two parallel lines m and n

In the above diagram, these $\angle 1=\angle 3, \angle 2=\angle 4$ are the corresponding angles, and $\angle 1=\angle 2$ are alternate interior angles.

Proof of Corresponding and Alternate Theorems

Alternate Angles Theorem

Given: Lines $p\parallel q$

To prove: $\angle 2 = \angle 7$ and $\angle 3 = \angle 6$

Proof of Alternate Angle Theorem

Proof: Let us assume that $p$ and $q$ are two parallel lines and $t$ is the transversal that intersects parallel lines $p$ and $q$. We know that if a transversal intersects any two parallel lines, the corresponding angles and vertically opposite angles are equal.

Therefore,

$\Rightarrow \angle 1 = \angle 3 \quad \ldots$ (i) (Corresponding angles)

$\Rightarrow \angle 1 = \angle 6$....(ii) (Vertically opposite angles)

From equations (i) and (ii), we get

$\Rightarrow \angle 3 = \angle 6 \quad$ (Alternate interior angles)

Similarly,

$\Rightarrow \angle 2 = \angle 7$

Hence, the proof of the Alternate Angle Theorem.

Corresponding Angles Theorem

Proof of Corresponding angle Theorem

To Prove:

$\angle 1 = \angle 5,\angle 3 = \angle 6,\angle 4 = \angle 7,\angle 2 = \angle 8$

Given, PQ and RS are the two parallel lines intersected by the transversal IJ.

Now, if $PQ\parallel RS$, then by the corresponding angles theorem, we can write

$\angle 1 = \angle 5$

$\angle 3 = \angle 6$

$\angle 4 = \angle 7$

$\angle 2 = \angle 8$

Remember: The theorem can only be proved when parallel lines are given.

Thus, this theorem is true without proof.

Limitations of Corresponding and Alternate Angle Theorems

• Both the theorems are applicable only if transversal lines intersect the parallel lines.

• They are not applicable in the case of nonparallel lines as the nonparallel lines do not form corresponding and alternate angles.

Applications of Corresponding and Alternate Angle Theorems

• Both Theorems are applicable in geometry to solve questions related to corresponding angles and alternate angles.

• Rubik's Cube, railway tracks, and opening and shutting of geometry boxes are examples of corresponding angles.

Solved Examples

1. Find the value of angle $x$ in the given figure if the two lines are parallel and they are crossed by a transversal.

Given figure to find angle x

Ans:

By the alternate interior angles theorem,

$x$ and $20^{\circ}$ are the alternate interior angles. Hence, they are equal.

Therefore,

$\Rightarrow x = 20^{\circ}$.

2. The two corresponding angles are given to be $9 x+10$ and 64. What is the value of $x$?

Ans: The two given corresponding angles are congruent.

$9 x+10= 64 \\$

$\Rightarrow 9 x=64-10 \\$

$\Rightarrow 9 x=54 \\$

$\Rightarrow x=6$

3. The values of two corresponding angles $\angle 2=5 x+6$ and $\angle 6=3 x+18$. Solve for the value of $x$.

Ans: As they are corresponding angles and the lines are said to be parallel in nature, then they should be congruent.

Equate the given expressions $\angle 2=5 x+2$ and $\angle 6=3 x+10$ and find the value of $x$.

$5 x+6=3 x+18$

$\Rightarrow 5 x-3 x=18-6$

$\Rightarrow 2 x=12$

$\Rightarrow x=\dfrac{12}{2}$

$\Rightarrow x=6$

Important Formulas to Remember

• Corresponding angles theorem: If two parallel lines are intersected by transversal lines, then the corresponding angles are equal.

• Alternate angles theorem: If two parallel lines are intersected by transversal lines, then alternate angles are equal.

Important Points to Remember

• Corresponding Angles are formed on the same side of parallel lines intersected by a transversal line.

• Alternate angles are formed on opposite sides of parallel lines intersected by a transversal line.