Alternate Angles

What are Alternate Angles?

In mathematical geometry, alternate angles are one of the most intriguing topics for students. Typically, the alternate angles can be considered as pairs of non-adjacent angles on either side of the transversal line. In an effort to understand alternate angles, here is all the information you need to know thoroughly.

What are Alternate Angles?

When a straight line intersects parallel lines, which are two or more in number then the lines are said to be transversal lines. In cases where this transverse line cuts any coplanar line, it leads to the formation of angles which are known as exterior or interior alternate angles. 

What are Alternate Interior Angles?

These are those angles that are situated between two intersecting lines. The alternate interior angles are generally on the opposite sides but in the interior of the transversal lines. Angles 3 and 6, as well as angles 5 and 4 in the below-given figure, are classic examples of alternate interior angles.

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What are Alternate Exterior Angles?

Just like the alternate interior angles, then alternate exterior angles are also located on the opposite sides of the transversal line but the only difference being that they are located on the exterior of the transverse. A classic example of the alternate exterior angle is shown in the below-mentioned figure where angles 1 and 8 along with angles 2 and 7 are known to be the alternate exterior angle. 

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All these angle relationships are formed only when two lines intersect each other at one point and under no other situation. The vertical angles, however, are an exception. With the exception of vertical angles, all of these relationships can only be formed when two lines are intersected by a transversal.

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Determining Angle Relationships

With so many similarities, people often end up wondering about the dynamic relationship between the angles formed by intersecting and transverse lines and the key factors for determining it. For this, all that needs to be done is to raise three basic questions:

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The first question is, if the angles are situated at the same position at both points of intersection?

This simply means to identify where the angle is lying. Whether it is in the upper left, upper right, lower left, or lower right corners of the intersections. If you determine that the angles are located at identical positions then these angles are said to be corresponding and your work here is done. If in case, these angles are not corresponding to each other then comes the time to ask the second question.

The second question is whether the angles are situated on the same or the opposite side of the transversal line. 

If the angles are positioned on the same side of the transverse line then the angles are said to be consecutive to each. In the case of angles on the opposite side of the transverse, the angles are known to be alternate. 

Alternate angles are always situated on the sides of the transversal which intersects the parallel line and these angles can be either on both the inner sides or both the outer ones of the parallel lines. Whereas, angles situated on the identical sides of the transversal wherein one angle is formed on the inside and the other on the outside of parallel lines are known to be corresponding angles. 

Alternate Angles Theorem

Alternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent.

To prove:

When a transversal cuts two parallel lines then alternate interior angles are formed which are equal. 

Proof:

Let us suppose that PQ and RS are the two lines that are running parallel to each other. These parallel lines PQ and RS are intersected by another transversal line LM. This line of intersection has further formed a few angles which are W, X, Y, Z.

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∠W + ∠Z = 180° (PQ is the straight line)—-(1)

∠X + ∠Z = 180° (LM is the straight line)—-(2)

So, from (1) and (2), we get

∠W = ∠X

Similarly

∠W + ∠Z = 180° (RS is the straight line)—-(3)

∠W + ∠Y = 180° (LM is the straight line)—-(4)

So, from (3) and (4), we get

∠Z = ∠Y

Therefore, it proved that alternate interior angles are congruent to each other. 

Hence, proved.

Solved Example

Two Parallel lines AB and CD are intersected by a transverse line. If m/_2 and m/_7

are denoted by 2x-40 and 1/2x+20 respectively, find the degree of /_5

Angle 2 and angle 7 are alternate exterior angles. And we know that angles which are alternate exterior are always congruent therefore, the measures of both these angles would be the same. Once this is determined, solve the question for “X”. 

2x-40=½ x+20

 1.5x-40+20 

1.5x+60 x+40

Plug in x to find m/_7

 m/_7= ½ x+20=½(40)+20=40 degree

which means that they are equal to a total of 180 degrees. Use this basic understanding to form an equation and solve for “x”. 

m/_5:

 m/_7+m/_5=180 degree  

40degree+ m/_5=180 degree

 m/_5=140 degree