Alternate Interior Angles: Definition, Theorem & More

Alternate Interior Angles: Definition, Theorem & More

Alternate interior angle is the angles formed when a transversal intersects two coplanar lines. They lie on the inner side of the parallel lines but the opposite sides of the transversal. The transversal crosses through the two lines which are Coplanar at separate points. These angles represent whether the two given lines are parallel to each other or not. If these angles are equal to each other then the lines crossed by the transversal are parallel.

An angle is formed when two rays, a line with one endpoint, meet at one point called a vertex. The angle is formed by the distance between the two rays. Angles in geometry are often referred to using the angle symbol so angle A would be written as angle A or When a line (called a transversal) intersects a pair of lines, AIAs are formed on opposite sides of the transversal. If the pair of lines are parallel then the alternate interior angles are equal to each other.

Alternate Interior Angles 

transversal line is a line that crosses or passes through two other lines. Sometimes, the two other lines are parallel, and the transversal passes through both lines at the same angle. The two other lines don’t necessarily have to be parallel in order for a transversal to cross them.

Angle

Angle

straight angle, also called a flat angle, is formed by a straight line. The measure of this angle is 180 degrees. A straight angle can also be formed by two or more angles that sum to 180 degrees. Here, angle 1 + angle 2 = 180.

Parallel lines are two lines on a two-dimensional plane that never meet or cross. When a transversal passes through parallel lines, there are special properties about the angles that are formed that do not occur when the lines are not parallel. Notice the arrows on lines m and n towards the left. These arrows indicate that lines m and n are parallel.

Alternate Interior Angles

AIAs are formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are AIAs. Notice the pairs of blue and pink angles.

Alternate Interior Angles Theorem/Proof

The theorem states that if a transversal crosses the set of parallel lines, the alternate interior angles are congruent.

Given: a//b

To prove: ∠4 = ∠5 and ∠3 = ∠6

Proof: Suppose a and b are two parallel lines and l is the transversal which intersects a and b at point P and Q. See the figure.

parallel lines and angles

parallel lines and angles

From the properties of the parallel line, we know if a transversal cuts any two parallel lines, the corresponding angles and vertically opposite angles are equal to each other. Therefore,

∠2 = ∠5            (i))(Corresponding angles)

∠2 = ∠4            (ii)(Vertically opposite angles)

From eq.(i) and (ii), we get

∠4 = ∠5 (Alternate interior angles)

Similarly,

∠3 = ∠6

Hence, it is proved.

Alternate Interior Angles

Alternate Interior Angles

Alternate Interior Angles Properties

  • These angles are congruent.
  • Some of the angles formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
  • In the case of non – parallel lines, alternate interior angles don’t have any specific properties.

What is the AIA?

Alternate interior angles are congruent. Formally, alternate interior angles are two interior angles that lie on different parallel lines and on opposite sides of a transversal.

Does AIA add up to 180?

Alternate angles form a ‘Z’ shape and are sometimes called Z angles. d and f are interior angles. These add up to 180 degrees (e and c are also interiors). Any two angles that add up to 180 degrees are known as supplementary angles.

What is the difference between the exterior and AIA?

When two lines are crossed by a transversal, the opposite angle pairs on the outside of the lines are an alternate exterior angle. One way to identify an alternate exterior angle is to see that they are the vertical angles of the alternate interior angle. An alternate exterior angle is equal to one another.

What are alternate angles in maths?

Alternate-angle. The word ‘alternate’ is usually used with pairs of angles, to indicate that each is on opposite sides of a line. In the figure below, the two angles are called alternate angles because they are on opposite sides of the sloping transversal line.

What are alternate angles in parallel lines?

If two parallel lines are cut by a transversal, then the alternate interior angle is congruent. If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. Alternate Exterior Angles: The word “alternate” means “alternating sides” of the transversal.

Do AIA equal to each other?

The alternate interior angle is formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are alternate interior angles. The theorem says that when the lines are parallel, the alternate interior angle is equal.

What are AIAs examples?

When two lines are crossed by another line (called the Transversal): Alternate Interior Angle is a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal. In this example, these are two pairs of Alternate Interior Angles: c and f.

What are alternate and corresponding angles?

Corresponding angles are at the same location on points of intersection. Next, we have the AIAs. Located between the two intersected lines, these angles are on opposite sides of the transversal.

How do you prove two lines are parallel without angles?

If two lines have a transversal that forms alternative interior angles that are congruent, then the two lines are parallel. If two lines have a transversal that forms corresponding angles that are congruent, then the two lines are parallel.

How do you prove that lines are parallel using coordinates?

When two straight lines are plotted on the coordinate plane, we can tell if they are parallel from the slope, of each line. If the slopes are the same then the lines are parallel. In the figure above, there are two lines that are determined by given points.

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