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.
