When a transversal intersects two coplanar lines, alternate interior angles are formed. Rather than lying on the inner side of the parallel lines, they lie on the corresponding side of the transversal line. There are two parallel lines which are Coplanar at separate points and hence the transversal crosses those lines. According to these angles, the two lines are parallel or not to each other depending on whether they are straight or curved. When these angles are equal to one another, then the lines that are crossed by the transversal are parallel. We will discuss in this article what are alternate interior angles, the theorem statements and proofs based on alternate interior angles, what is a co-interior angle, and many examples that are solved.
Alternate Interior Angles:
An alternate interior angle is formed when two parallel lines are crossed by a transversal, but on the opposite sides. These angles are always the same. Another way of putting this is that they are inverse angles. Alternative interior angles can be used to determine whether or not two lines are parallel. The lines that are crossed by transversals are said to be parallel if these angles are equal.
The alternate interior angles are shown in the figure below. The two parallel lines AB and CD are crossed by a transversal.The pairs of alternate interior angles in the above figure are determined by the alternate interior angle theorem:
- ∠4 and ∠6
- ∠3 and ∠5
We will briefly discuss alternate exterior angles and how they differ from alternate interior angles.
The Alternate Interior Angles/Converse Interior Angles Theorem:
The theorem states that ” if a transversal-crosses a parallel set of lines, the alternate interior angles are congruent”.
To prove: ∠4 = ∠5 and ∠3 = ∠6
Let a and b be two parallel lines, and let l be a transversal which intersects a and b at points P and Q. This is illustrated in Figure 1.
We know that if a transversal cuts any two parallel lines, the corresponding angles and the vertically opposite angles will be equal. Hence,
∠2 = ∠5 =(i) [Corresponding angles]
∠2 = ∠4 = (ii) [Vertically opposite angles]
From eq.(i) and (ii), we get;
∠4 = ∠5 [Alternate interior angles]
∠3 = ∠6
Hence, it is proved.
Antithesis of Theorem:
As long as the interior angles produced by the transversal line on two lines are congruent, then these two lines are parallel to each other.
Given: ∠4 = ∠5 and ∠3 = ∠6
To prove: a//b
Proof: Since ∠2 = ∠4 [Vertically opposite angles]
So, we can write,
∠2 = ∠5, which are corresponding angles.
Therefore, a is parallel to b.
Alternate Interior Angles Have the Same Shape?
When a transversal passes through two lines, it forms alternate interior angles. The angles that are formed on opposite sides of the transversal line and within the two lines are known as alternate interior angles. It is a theorem that states that when the lines are parallel, the alternate interior angles will be equal.
Which Angle Does the Alternate Segment Have?
It is also known as the tangent-chord theorem. This theorem states that in any circle, the angle between a chord and a tangent passing through one of the endpoints of the chord is equal to the angle formed by the alternate segment. As you can see in the above diagram, angles of the same color are equal to each other.
Alternate Angles Congruent with The Interior:
According to the Interior Angles Theorem, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
A theorem is a statement that has been proven to be true or an idea that has been accepted as correct. This theorem also has a proven converse, which is that if two lines are cut by a transversal and the alternate interior angles are congruent, then they are parallel.
These theorems can be used to solve problems in geometry and to find missing information. This diagram shows which pairs are equal and alternate interior. Notice that the lines are parallel.