Video Transcript
In this video, we’ll learn how to
prove certain geometric properties using deductive proof. Now, lots of these geometric
properties will be properties that we’re probably already very familiar with and
ones that we use all the time in geometry. However, what we want to do here is
learn how to prove these properties and then use these to prove even more
properties. Let’s have a look at some of these
now.
The first one is the angles on a
straight line. We can recall that the angle
measures on a straight line sum to 180 degrees. And we can use this fact in many
other proofs. Let’s take, for example, how we
might use this fact to find and prove the sum of the angle measures about a
point. To do this, we can consider an
arrangement of lines as shown. We have the rays 𝑂𝐴, 𝑂𝐵, 𝑂𝐶,
and 𝑂𝐷. We want to calculate the sum of the
angles about a point 𝑂.
Let’s construct the line 𝐴𝑂,
which passes through the point 𝐸. And since we know that the angle
measures on a straight line sum to 180 degrees, we can write that the measure of
angle 𝐴𝑂𝐵 plus the measure of angle 𝐵𝑂𝐶 plus the measure of angle 𝐶𝑂𝐸 is
equal to 180 degrees. And we can do the same on the other
side of the line 𝐴𝑂. The measure of angle 𝐴𝑂𝐷 plus
the measure of angle 𝐷𝑂𝐸 equals 180 degrees, since these also lie on a straight
line.
And since we want to work out the
total sum of the angles about point 𝑂, we can finish by adding all the angles. We can take the measures of the
first three angles — 𝐴𝑂𝐵, 𝐵𝑂𝐶, and 𝐶𝑂𝐸 — and add them to the measure of the
next two angles — 𝐴𝑂𝐷 and 𝐷𝑂𝐸. And since we know that both sets of
angles must add to 180 degrees, then their total must be 360 degrees. And so we have proved a familiar
geometric property by using the property of the angles on a straight line. The sum of the measures of the
angles about a point is 360 degrees.
Now let’s see the next definition
and property. These are vertically opposite
angles, defined as the angles created when two straight lines intersect. We recall that if two straight
lines intersect, then the vertically opposite angles, or sometimes simply the
opposite angles, are equal in measure. And yes, we might already know this
property. But this is the important question:
how do we prove that vertically opposite angles are equal? Well, let’s see how in the
following example.
Two straight lines, 𝐴𝐵 and
𝐶𝐷, intersect at point 𝐸. Fill in the blank. If the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐶
are adjacent angles, where the union of rays 𝐸𝐶 and 𝐸𝐷 equals the line
segment 𝐶𝐷, then the measure of angle 𝐴𝐸𝐶 plus the measure of angle 𝐴𝐸𝐷
equals what. Fill in the blank. If the angles 𝐴𝐸𝐶 and 𝐶𝐸𝐵
are adjacent angles, where the union of rays 𝐸𝐴 and 𝐸𝐵 equals the line
segment 𝐴𝐵, then the measure of angle 𝐴𝐸𝐶 plus the measure of angle 𝐶𝐸𝐵
equals what. True or False: We deduce from
the two parts above that the measure of angle 𝐴𝐸𝐷 equals the measure of angle
𝐶𝐸𝐵.
Let’s begin this question by
drawing the two given lines, 𝐴𝐵 and 𝐶𝐷, which intersect at a point 𝐸. Notice that we could have drawn
any different diagram of the lines 𝐴𝐵 and 𝐶𝐷 that intersect at point 𝐸, so
long as it shows that important line and intersection information. We would still be able to use
any such diagram to answer the questions.
So let’s use the first diagram
and look at the first part of this question. Here, we need to first identify
the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐶. The second part of this
sentence, which tells us that the union of rays 𝐸𝐶 and 𝐸𝐷 is the line
segment 𝐶𝐷, is really stating the fact that these lines form one straight-line
segment. And what do we know about the
angles on a straight line? Well, the angle measures on a
straight line sum to 180 degrees. And so, the measure of angle
𝐴𝐸𝐷 plus the measure of angle 𝐴𝐸𝐶 is equal to 180 degrees. And that’s the first missing
blank completed.
Let’s look at the second part
of the question. This time, we’re looking at the
angles 𝐴𝐸𝐶 and 𝐶𝐸𝐵. Once again, we’re told that the
rays 𝐸𝐴 and 𝐸𝐵 form one straight-line segment, 𝐴𝐵. And we know that the angle
measures on a straight line sum to 180 degrees. So these two angle measures of
𝐴𝐸𝐶 and 𝐶𝐸𝐵 will also sum to 180 degrees. So now we have answered the
second part of this question.
Let’s look at the final
part. In this part, we are
considering the angle measures of 𝐴𝐸𝐷 and 𝐶𝐸𝐵. To help us with this, we’ll
consider what we discovered in parts one and two. In the first part, we
recognized that the measures of angles 𝐴𝐸𝐶 and 𝐴𝐸𝐷 added to give 180
degrees. Let’s label the measure of
angle 𝐴𝐸𝐷 as 𝑥 degrees and the measure of angle 𝐴𝐸𝐶 as 𝑦 degrees. In the second part of the
question, we identified another pair of angle measures that added to 180
degrees. And since 𝑥 degrees plus 𝑦
degrees equals 180 degrees, then we can say that the measure of angle 𝐶𝐸𝐵 is
also 𝑥 degrees.
So, we can say that the
statement that the measure of angle 𝐴𝐸𝐷 equals the measure of angle 𝐶𝐸𝐵 is
true. And in fact, what we have here
is a proof that vertically opposite angles are equal. We could even have continued in
this example to demonstrate that the measure of angle 𝐴𝐸𝐶 equals the measure
of angle 𝐷𝐸𝐵. These vertically opposite
angles will both be 𝑦 degrees.
When we are proving geometric
properties, another useful tool to have is to be familiar with the properties of the
angles created in parallel lines. We can recall these properties
now. When a pair of parallel lines is
intersected by another line, known as a transversal, it creates pairs of congruent
or supplementary angle measures. Firstly, we have alternate angles,
which are equal in measure. We have corresponding angles, which
are also equal in measure. And we have supplementary interior
angles. The measures of these angles sum to
180 degrees. We’ll now see how we can use these
properties in the following example.
True or False: A straight line
that is perpendicular to one of two parallel lines is also perpendicular to the
other.
The best way to understand
fully what is asked here is to start by drawing a diagram. Let’s start with the straight
line. We also know that there are two
parallel lines. And one of the two parallel
lines is perpendicular to the straight line. It might also be useful if we
label the lines so that we can start to use these as part of a proof. So, let’s say that the two
parallel lines are the lines 𝐴𝐵 and 𝐶𝐷 and the line which is perpendicular
to line 𝐴𝐵 is the line 𝐸𝐹. We can also label the point
where line 𝐴𝐵 and 𝐸𝐹 intersect as 𝑃 and where line 𝐶𝐷 and 𝐸𝐹 intersect
as point 𝑄. If we wanted to use some
mathematical notation, we could write our facts like this.
We now need to work out if this
statement in the question is true. Is the other line, which we’ve
called line 𝐶𝐷, also perpendicular to line 𝐸𝐹? Since we know that we have this
relationship where the two lines are perpendicular, we can write that the
measure of angle 𝐸𝑃𝐵 is 90 degrees. Then, we can use the properties
of parallel lines to help us with another fact. Angles 𝐸𝑃𝐵 and 𝐸𝑄𝐷 are
corresponding angles, and we know that corresponding angles are equal. Since both of these angles are
equal, they are both equal to 90 degrees. Therefore, line 𝐸𝐹 is also
perpendicular to line 𝐶𝐷. The statement in the question
is true.
And so, we have used our
knowledge of geometry to prove a geometrical fact. By using the properties of
parallel lines, we have proved that a straight line that is perpendicular to one
of two parallel lines is also perpendicular to the other.
Very commonly in geometry proofs
involving two-dimensional shapes, we need to apply the criteria for proving that two
triangles are congruent. Let’s recap these next.
The first criterion we could use to
prove that two triangles are congruent is the side-angle-side, or SAS, criterion,
which states that two triangles are congruent if they have two sides that are
congruent and the included angle is congruent, next the angle-side-angle or ASA
criterion, which states that two triangles are congruent if they have two angles
that are congruent and the included side is congruent, thirdly the SSS criterion,
that two triangles are congruent if they have all three sides congruent, and finally
the criterion that only applies in right triangles, the RHS criterion, which states
that two triangles are congruent if they both have a right angle and the hypotenuse
and one other side are congruent.
As previously mentioned, we often
use these criteria to help us prove further properties. In the next question, we’ll see how
we can use congruent triangles to demonstrate one of the properties of a kite.
In the given figure, use the
properties of congruent triangles to find the measure of angle 𝐵𝐶𝐷.
So, let’s begin by looking at
this figure and identifying any key information from the markings on it. Firstly, we have two pairs of
congruent line segments marked. Sides 𝐴𝐵 and 𝐴𝐷 are
congruent, and sides 𝐶𝐵 and 𝐶𝐷 are congruent. We also have the angle measure
of angle 𝐴𝐶𝐷 given as 29 degrees. Since we are told to use the
properties of congruent triangles, let’s identify two triangles we may be able
to use.
We have triangle 𝐴𝐵𝐶 and
triangle 𝐴𝐷𝐶. These two triangles share a
common side of 𝐴𝐶. So this length will be equal in
both triangles. And so, we have in fact
recognized that there are three pairs of congruent sides. We can then write that triangle
𝐴𝐵𝐶 is congruent to triangle 𝐴𝐷𝐶 by the SSS congruency criterion. So, all corresponding pairs of
sides and angles in each triangle are congruent.
We can then identify a pair of
congruent angles. Given that angles 𝐴𝐶𝐵 and
𝐴𝐶𝐷 are corresponding, these will both be 29 degrees. But we are asked to find the
measure of angle 𝐵𝐶𝐷. Since angle 𝐵𝐶𝐷 is comprised
of angles 𝐴𝐶𝐵 and 𝐴𝐶𝐷, we need to add their measures of 29 degrees each,
which gives us a final answer of 58 degrees.
In this example, we find an
unknown angle in a kite. But there is also another
property proved here which may not be immediately obvious. And that is that the longer
diagonal of a kite bisects the angles at the vertices on this diagonal. Although we used specific
values for the measures of angles 𝐴𝐶𝐵 and 𝐴𝐶𝐷, we could just have easily
had a general angle of 𝑥 degrees for each. Since a kite must have two
pairs of congruent sides, we will always be able to create two congruent
triangles within any kite. And so this angle of two 𝑥
degrees at the end of the longer diagonal in the kite will always be bisected by
this diagonal. And because of the congruency
of the triangles, we can also say the same for the angle at the other end of the
longer diagonal. It would also be bisected.
We will now summarize the key
points of this video. We saw throughout this video how we
can use geometric properties to prove further geometric properties as part of a
deductive proof. Many of the important parts of our
proofs come from properties including the angle measures on a straight line, the
angles in parallel lines, and congruent triangles. We used geometric properties to
prove that the sum of the angles about a point is 360 degrees. If two straight lines intersect,
then the vertically opposite angles are equal in measure. And we proved that the longer
diagonal of a kite bisects the angles at the vertices on this diagonal.
