Lesson Video: Parallel Lines and Transversals: Angle Relationships Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to name and identify angle pairs formed by parallel lines and transversals and recognize their relationships to find a missing angle. Before looking at parallel lines, we will recall some other angle properties and relationships.

Let’s begin by looking at vertically opposite angles. Vertically opposite angles are two angles between two second lines that share a vertex. The phrase “second lines” means two lines that cross each other. Looking more closely at the four angles shown, we can see that we have two pairs of equal angles. Angles 𝑎 and 𝑐 are vertically opposite and angles 𝑏 and 𝑑 are also vertically opposite. This means that the adjacent angles will sum to 180 degrees. For example, 𝑎 plus 𝑏 is equal to 180 degrees and 𝑐 plus 𝑑 is equal to 180 degrees. This is because the sum of any two angles on a straight line is equal to 180.

All four angles shown will sum to 360 degrees. This is because angles in a circle or at a point sum to 360. Angle 𝑎 plus angle 𝑏 plus angle 𝑐 plus angle 𝑑 equals 360 degrees. We will now look at the case of two parallel lines crossed by a third line.

In this diagram, we have two parallel lines, L one and L two, and a third transversal line, L three, that cuts them. We have eight angles created. And we recognize there are four pairs of vertically opposite angles, 𝑎 and 𝑐, 𝑏 and 𝑑, 𝑒 and 𝑔, and 𝑓 and ℎ.

Because the lines L one and L two are parallel, the two sets of four angles between L three and L one and between L three and L two are congruent. This means that angles 𝑎, 𝑐, 𝑒, and 𝑔 are equal. Likewise, angles 𝑏, 𝑑, 𝑓, and ℎ are equal. These facts lead us to three relationships we can use to solve problems when dealing with parallel lines. Our first pair of congruent angles are called corresponding or F angles. These are in the corresponding position with respect to the transversal line, L three, and one of the parallel lines, L one or L two.

Secondly, we have alternate angles, also known as “zee” or “zed” angles. These are formed by a transversal line, L three, cutting two parallel lines, L one and L two, that are on either side of L three and between L one and L two.

Finally, we have cointerior or C angles. Unlike corresponding and alternate angles which are congruent, cointerior or C angles sum to 180 degrees. This leads us onto the parallel lines theorem, which states that when two parallel lines are cut by a transversal line, then the pairs of corresponding angles are congruent and the pairs of alternate angles are congruent. We will now look at some questions to see how we can apply these relationships.

In the figure, 𝐸𝑁 intersects 𝐴𝐵 and 𝐶𝐷 at 𝑀 and 𝐹, respectively. Find the measure of angle 𝐸𝐹𝐶.

The lines through 𝐴𝐵 and 𝐶𝐷 are parallel as indicated on the diagram. The line 𝐸𝑁 is a transversal line that cuts the two parallel lines. We have been asked to work out the value of angle 𝐸𝐹𝐶. In order to answer this question, we will use our angle properties relating to parallel lines.

We know that vertically opposite angles are equal. This means that angle 𝐸𝑀𝐵 is equal to angle 𝐴𝑀𝐹. Both of these are equal to 84 degrees. We also know that cointerior or supplementary angles sum to 180 degrees. These are often referred to as C angles, as shown in the diagram. Angles 𝐴𝑀𝐹 and 𝐸𝐹𝐶 must sum to 180 degrees. This means that 84 plus angle 𝐸𝐹𝐶 is equal to 180. Subtracting 84 from both sides of this equation gives us angle 𝐸𝐹𝐶 is equal to 96. The measure of angle 𝐸𝐹𝐶 is equal to 96 degrees.

We will now look at another question involving parallel lines.

Find the measure of angle 𝐶.

We can see from the diagram that the line 𝐴𝐵 is parallel to the line 𝐶𝐷. In this question, we need to calculate the measure of angle 𝐶. We begin by looking at the point 𝐴, noting that angles at a point or in a circle sum to 360 degrees. If we let the missing angle be 𝑥, then 𝑥 plus 123 plus 132 is equal to 360. Simplifying this gives us 𝑥 plus 255 is equal to 360. Subtracting 255 from both sides of this equation gives us 𝑥 is equal to 105. The missing angle at point 𝐴 is 105 degrees.

As mentioned previously, lines 𝐴𝐵 and 𝐶𝐷 are parallel. The line 𝐴𝐶 creates two cointerior or supplementary angles. As these sum to 180 degrees, the angle 𝑦 at point 𝐶 plus 105 must equal 180. Subtracting 105 from both sides of this equation gives us 𝑦 is equal to 75. We can therefore conclude that the measure of angle 𝐶 is equal to 75 degrees.

Our third question will involve parallel lines and also a quadrilateral.

In the figure, 𝐶𝐷 and 𝐵𝐸 are parallel. Find the measure of angle 𝐴𝐵𝐸.

We are told in the question that the lines 𝐶𝐷 and 𝐵𝐸 are parallel. We’re asked to calculate the size of angle 𝐴𝐵𝐸 denoted by the letter 𝑥. We can see from the diagram that 𝐴𝐵𝐶𝐷 is a quadrilateral, a four-sided shape. The angles in any quadrilateral sum to 360 degrees. This means that the sum of the missing angle 𝑦, 90 degrees, 131 degrees, 69 degrees must equal 360 degrees. Simplifying the left-hand side gives us 𝑦 plus 290 is equal to 360. Subtracting 290 from both sides gives us 𝑦 is equal to 70. The missing angle in the quadrilateral is 70 degrees.

We can now use the fact that cointerior or supplementary angles sum to 180 degrees. These are also sometimes known as C angles. In this question, 70 plus 69 plus 𝑥 must equal 180. This can be simplified to 𝑥 plus 139 is equal to 180. Subtracting 139 from both sides of this equation gives us 𝑥 is equal to 41. We can therefore conclude that the measure of angle 𝐴𝐵𝐸 is 41 degrees.

Our next question also involves alternate angles.

From the information in the figure below, find the measure of angle 𝐴𝐸𝐶.

Angle 𝐴𝐸𝐶 is shown in the diagram. This can be split into the sum of two other angles, angle 𝐴𝐸𝐹 and angle 𝐶𝐸𝐹. We can use our angle properties involving parallel lines to calculate these two values. The lines 𝐴𝐵, 𝐸𝐹, and 𝐶𝐷 are parallel as indicated on the diagram. We know that alternate angles are congruent. These are sometimes referred to as “zee” or “zed” angles. This means that angle 𝐵𝐴𝐸 is equal to angle 𝐴𝐸𝐹. They’re both equal to 92 degrees.

Cointerior or supplementary angles sum to 180 degrees. These are often known as C angles. In this question, angle 𝐶𝐸𝐹 plus 131 degrees is equal to 180 degrees. This means that our missing angle 𝑦, angle 𝐶𝐸𝐹, is equal to 49 degrees. We can therefore calculate the measure of angle 𝐴𝐸𝐶 by adding 92 degrees and 49 degrees. This is equal to 141 degrees.

The final question we will look at involves angle properties of a parallelogram.

Find the measure of angle 𝐹𝐺𝐸.

The diagram shows a parallelogram, where lines 𝐸𝐹 and 𝐺𝐻 are parallel. The sides 𝐸𝐻 and 𝐹𝐺 are also parallel. We can therefore use our angle properties involving parallel lines to find the measure of angle 𝐹𝐺𝐸. We begin by recalling that alternate angles are congruent. These are often referred to as “zed” or “zee” angles. In this question, angle 𝐻𝐺𝐸 is equal to angle 𝐺𝐸𝐹. They are both equal to 31 degrees.

𝐸𝐹𝐺 is a triangle. And we know that angles in a triangle sum to 180 degrees. This means that 𝑥 plus 31 plus 106 is equal to 180. Simplifying this gives us 𝑥 plus 137 is equal to 180. Finally, subtracting 137 from both sides gives us 𝑥 equals 43. We can therefore conclude that the measure of angle 𝐹𝐺𝐸 is equal to 43 degrees.

We will now finish this video by summarizing the key points.

We have three main angle properties involving parallel lines. Firstly, corresponding or F angles are congruent. Alternate angles are also congruent. These are known as “zee” or “zed” angles. Cointerior or supplementary angles sum to 180 degrees. These are also known as C angles. When dealing with any problem involving parallel lines, we also need to be aware of our properties involving triangles, quadrilaterals, straight lines, and vertically opposite angles.

The parallel lines theorem states that when parallel lines are cut by a transversal, then the pairs of corresponding and alternate angles are congruent. This is shown in the diagram where lines L one and L two are parallel and L three is the transversal. The angles 𝑎, 𝑐, 𝑒, and 𝑔 are equal, as are the angles 𝑏, 𝑑, 𝑓, and ℎ. The four angles at the intersection of L one and L three are congruent to those angles at the intersection of L two and L three.