Alternate Interior Angles are formed when a transversal line meets two or more parallel lines. In simple words, Alternate interior angles are a pair of angles formed when a transversal intersects a pair of lines. Alternate Interior Angles are on opposing sides of the transversal and inside the parallel lines. Alternate interior angles are congruent, which means they have the same measure or size. This is a fundamental property of parallel lines.
For example, if angles A and B are alternative internal angles, then they are congruent. This property of Alternate Interior Angles is crucial in geometry and is utilized in many proofs and computations involving parallel lines. Understanding alternate interior angles aids in the solution of issues involving angles and parallel lines, making them a critical idea in geometry.
In this article, we will discuss the concept of Alternate Interior Angles in detail including its definition, example, and properties. We will also learn about the Alternate Interior Angle Theorem and its converse and their proof as well.
What are Alternate Interior Angles?
When two lines are crossed by a third line, alternate internal angles are generated. The third line is known as the transversal line. When two parallel or non-parallel lines are crossed by a transversal line, alternate interior angles are always created. The angles are on opposing sides of the transversal line and on the inside of parallel or non-parallel lines. This is why they are termed alternative interior angles: they are opposed and on the inside of the two lines.
Alternate Interior Angles Definition
The angles created on opposing sides of the transversal are known as alternate internal angles. In other words, when two parallel lines are crossed by a transversal, eight angles are generated. The alternate interior angles are those that are on the inner side of the parallel lines but on the opposite sides of the transversal.
Alternate Interior Angles Examples
Let’s consider a pair of parallel line and a transversal as shown in the figure below:
The inner angles are angles ∠3, ∠4, ∠5, and ∠6. Out of these ∠3 & ∠6 and ∠4 & ∠5 are the examples of Alternate Interior Angles.
Note: Angles ∠3 and ∠5, ∠4 and ∠6, ∠1 and ∠7, and ∠2 and ∠8 are Alternates Angles.
Properties of Alternate Interior Angles
There are various properties of Alternate Interior Angles, some of these properties are:
- Alternate Interior angles that are opposite one another are congruent (measure the same).
- Consecutive interior angles are alternate interior angles on the same side of the transversal line.
- The total of the angles created on the same side of the transversal that are within the two parallel lines is always 180°.
- Alternate interior angles do not have any special qualities in the case of non-parallel lines.
Are Alternate Interior Angles Congruent?
Alternate internal angles are generated when a transversal intersects two parallel lines. These angles are positioned between the parallel lines on opposing sides of the transversal. Importantly, alternate internal angles are congruent, which means they have the same dimensions. This geometric characteristic has several uses, notably in mathematics and engineering. Understanding this notion allows you to calculate unknown angles and solve complicated geometry problems involving parallel lines and transversals. This foundational concept is critical in geometry, serving as the foundation for theorems and proofs in higher mathematical subjects.
Alternate Interior Angles Theorem
According to the theorem, “a transversal crosses the set of parallel lines if the alternate interior angles are congruent.”
Proof of Alternate Interior Angles Theorem
Given: l || m
To Prove: ∠4 = ∠5 and ∠3 = ∠6
Proof: Assume that t is the transversal that crosses l and m. l and m are two parallel lines. See the illustration below.
We know from the characteristics of the parallel line that the corresponding angles and vertically opposed angles are identical when a transversal intersects any two parallel lines. Therefore,
From the properties of the parallel line, we know if transversal cuts any two parallel lines, the corresponding angles and vertically opposite angles are equal to each other. Therefore,
∠1 = ∠5 . . . (i) [Corresponding Angles]
∠1 = ∠4 . . . (ii) [Vertically Opposite Angles]
From Equation (i) and Equation (ii), we get
∠4 = ∠5 [Alternate interior angles]
Similarly,
∠3 = ∠6
So, it is proven.
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Converse of Alternate Interior Angles Theorem
If two lines are crossed by a transversal and the alternate interior angles are congruent, the lines must be parallel, according to the converse of Alternate Interior Angles. This theorem is useful for finding parallel lines based on the equality of their alternate interior angles, and it has practical applications in geometric reasoning and problem solving.
Proof of Converse of Alternate Interior Angles Theorem
Converse of Alternate Interior angles that are opposite each other. According to the theorem, if two lines and a transversal form alternating interior angles that are equivalent, the lines are parallel.
Given: Transversal EFGH connects lines AB and CD, resulting in a pair of equal alternative E.
To Prove: AB||CD
Proof:
∠AFG = ∠FGD
But ∠AFG = ∠EFB (Vertically opposite angles)
⇒∠EFB = ∠FGD (corresponding angles)
⇒AB || CD
The lines are parallel because their respective angles, AB and CD, are equal. As a result, the reverse of the alternative interior angles theorem is demonstrated.
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Solved Examples of Alternate Interior Angles
Example 1: Demonstrate that the interior angles (6x + 82)° and (8x – 88)° are congruent.
Answer:
Interior alternate angles are equal, As a result,
(7x + 82) ° = (8x – 88) °
⇒ 7x + 82 = 8x – 88
⇒ x = 170
In the original formulations, substitute x.
⇒ (7x + 82) ° = 1272°.
⇒ (8x – 88)° = 1272°
As a result, (7x + 82) ° = (8x – 88)°.
Example 2: P and Q are on opposite lanes. They planned to construct a path that would cut at an angle between the two pathways. What is the value of ∠Q if ∠P = 85°?
Answer:
Because the two pathways are joined by a transversal, the alternative angles should be identical, suggesting that ∠P equals ∠Q.
As a result, ∠Q = ∠P = 85°.
Example 3: If the alternate interior angles are indicated as (7x + 55)° and (8x – 95)° in a given set of two parallel lines cut by a transversal, calculate the value of x and the true value of the alternate interior angles using the alternate interior angles theorem.
Answer:
Alternate interior angles are (7x + 55)° and (8x – 95)°. L1 and L2 are parallel, hence the two angles are congruent. So,
(7x + 55)° = (8x – 95)°
⇒ 7x + 55 = 8x – 95
⇒ -150 = -x
As a result, x = 150°.
By changing the value of x with the provided angles, we can determine the exact value of the angles.
(7x + 55)° = [7(150) + 55]° = 1105°
(8x – 95)°= [8(150) − 95]° = 1105°
Because they are opposite inside angles, they have the same value.
Example 4: Mark the interior angles and write their values on the accompanying picture. In the diagram, ∠3 = 95°and ∠5 = 105°.
Solution:
The various inside angles in the preceding image are ∠3, ∠4, ∠5, and ∠6. Only ∠3 and ∠5 are offered.
Because we already know that alternate inside angles are equivalent.
∠3 = ∠6 = 95°
∠4 = ∠5 = 105°
Alternate Interior Angles – FAQs
1. Define Alternate Interior Angles.
The angles created inside the two parallel lines when a transversal intersects them are known as the alternate interior angles, and they are equal to their alternate pairs.
2. Are Alternate Interior Angles Equal?
Yes, alternate internal angles are equal for parallel lines. When a transversal connects two parallel lines, the angles on opposing sides of the transversal have the same measure.
3. How Can Alternate Interior Angles Be Solved?
The alternate interior angles theorem states that if a transversal connects two parallel lines, the alternate interior angles are identical in size. If we know the measure of the corresponding alternate interior angle, we may use this theorem to determine the measure of the alternate interior angle.
4. Alternate Interior and Exterior Angles: What Is the Difference?
The angles that are between the interior of the two lines and have alternate interior angles are those with different vertices, which are on the alternate sides of the transversal. Alternate exterior angles, on the other hand, are those angles with distinct vertices that are located on the alternate sides of the transversal but are located on the outside of the two lines.
5. How Can Alternate Interior Angles Be Solved?
The alternate interior angles theorem states that if a transversal connects two parallel lines, the alternate interior angles are identical in size. If we know the measure of the corresponding alternate interior angle, we may use this theorem to determine the measure of the alternate interior angle.
6. Alternative Interior Angles: Supplementary or Congruent?
The alternate interior angles of two parallel lines are congruent or identical, according to the alternate interior angles theorem.
7. What are Same Side of Interior Angles?
Same-side interior angles are pairs of angles generated by two parallel lines and a transversal that are on the same side of the transversal. They add up to 180 degrees.
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