Alternate Interior Angles Definition
Alternate Interior Angles: 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.
Alternate Interior Angles TheoremA 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.
Alternate Interior Angles Definition
In the drawing below, angles 3 and 6 are alternate interior angles, as are angles 4 and 5. Alternate angles are congruent. Formally, alternate interior angles are two interior angles that lie on different parallel lines and on opposite sides of a transversal.
In geometry, pairs of angles can relate to each other in several ways.
In these lessons, we will learn
- alternate angles
- proof of the alternate interior angle theorem
- proof of the converse of the alternate interior angle theorem
- alternate exterior angles
- proof of the alternate exterior angle theorem
- proof of the converse of the alternate exterior angle theorem
What Are Alternate Interior Angles
The following figure shows some examples of alternate interior and alternate exteriors angle pairs. Scroll down the page if you need more examples and explanations about alternate angles and alternate exterior angles.
Alternate Interior Angles
When a line (called a transversal) intersects a pair of lines, alternate angles are formed on opposite sides of the transversal. If the pair of lines are parallel then the alternate angles are equal to each other.
The Alternate Angles Theorem states that
When two parallel lines are cut by a transversal, the resulting alternate angles are congruent.
How to identify Alternate Interior Angles?
One way to find the alternate angles is to draw a zigzag line on the diagram. In the above diagrams, d and e are alternate angles. Similarly, c and f are also alternate angles.
(Angles found in a Z-shaped figure)
Are Alternate Interior Angles Congruent
A 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.
These pairs are alternate angles. Alternate angles 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 alternate angles. Notice the pairs of blue and pink angles. 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.
Another way to think about alternate interior angles is by using the z-pattern. Notice that the pair of alternate interior angles makes a Z.
In this windowpane, angle a and angle b are alternate angles because they are on opposite sides of the transversal but inside the parallel lines.
Converse Of Alternate Interior Angles Theorem
The Alternate Interior theorem states, if two parallel lines are cut by a transversal, then the pairs of the alternate interior are congruent.
A theorem is a proven statement or an accepted idea that has been shown to be true. The converse of this theorem, which is basically the opposite, is also a proven statement: if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
These theorems can be used to solve problems in geometry and to find missing information. This diagram shows which pairs of angles are equal and alternate interior. Notice that the lines are parallel.
Next, notice that angle x and 58 form a straight angle. Since a straight angle measures 180 degrees, angle x + 58 = 180 and 180 – 58 = angle x, so angle x = 122.First, we need to identify a pair of alternate angles. Angle y and 58 are on opposite sides of the transversal and inside the parallel lines, so they must be alternate interior. Since the lines are parallel, angle y = 58.
Finally, angle x and angle z are alternate interiors, and we know that alternate angles are equal. So, angle x = 122 then angle z = 122.
Do alternate interior angles 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 alternate interior angles?
When two lines are crossed by a transversal, the opposite angle pairs on the outside of the lines are alternate exterior angles. … One way to identify alternate exterior angles is to see that they are the vertical angles of the alternate interior angles. Alternate exterior angles are equal to one another.