Video Transcript
In this video, weβll learn how to
prove certain geometric properties using deductive proof. Now, lots of these geometric
properties will be properties that weβre 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. Letβs 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. Letβs 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 ππ΄, ππ΅, ππΆ,
and ππ·. We want to calculate the sum of the
angles about a point π.
Letβs construct the line π΄π,
which passes through the point πΈ. And since we know that the angle
measures on a straight line sum to 180 degrees, we can write that the measure of
angle π΄ππ΅ plus the measure of angle π΅ππΆ plus the measure of angle πΆππΈ is
equal to 180 degrees. And we can do the same on the other
side of the line π΄π. The measure of angle π΄ππ· plus
the measure of angle π·ππΈ 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 π, we can finish by adding all the angles. We can take the measures of the
first three angles β π΄ππ΅, π΅ππΆ, and πΆππΈ β and add them to the measure of the
next two angles β π΄ππ· and π·ππΈ. 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 letβs 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, letβs see how in the
following example.
Two straight lines, π΄π΅ and
πΆπ·, intersect at point πΈ. Fill in the blank. If the angles π΄πΈπ· and π΄πΈπΆ
are adjacent angles, where the union of rays πΈπΆ and πΈπ· equals the line
segment πΆπ·, then the measure of angle π΄πΈπΆ plus the measure of angle π΄πΈπ·
equals what. Fill in the blank. If the angles π΄πΈπΆ and πΆπΈπ΅
are adjacent angles, where the union of rays πΈπ΄ and πΈπ΅ equals the line
segment π΄π΅, then the measure of angle π΄πΈπΆ plus the measure of angle πΆπΈπ΅
equals what. True or False: We deduce from
the two parts above that the measure of angle π΄πΈπ· equals the measure of angle
πΆπΈπ΅.
Letβs begin this question by
drawing the two given lines, π΄π΅ and πΆπ·, which intersect at a point πΈ. Notice that we could have drawn
any different diagram of the lines π΄π΅ and πΆπ· that intersect at point πΈ, 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 letβs use the first diagram
and look at the first part of this question. Here, we need to first identify
the angles π΄πΈπ· and π΄πΈπΆ. The second part of this
sentence, which tells us that the union of rays πΈπΆ and πΈπ· is the line
segment πΆπ·, 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
π΄πΈπ· plus the measure of angle π΄πΈπΆ is equal to 180 degrees. And thatβs the first missing
blank completed.
Letβs look at the second part
of the question. This time, weβre looking at the
angles π΄πΈπΆ and πΆπΈπ΅. Once again, weβre told that the
rays πΈπ΄ and πΈπ΅ form one straight-line segment, π΄π΅. And we know that the angle
measures on a straight line sum to 180 degrees. So these two angle measures of
π΄πΈπΆ and πΆπΈπ΅ will also sum to 180 degrees. So now we have answered the
second part of this question.
Letβs look at the final
part. In this part, we are
considering the angle measures of π΄πΈπ· and πΆπΈπ΅. To help us with this, weβll
consider what we discovered in parts one and two. In the first part, we
recognized that the measures of angles π΄πΈπΆ and π΄πΈπ· added to give 180
degrees. Letβs label the measure of
angle π΄πΈπ· as π₯ degrees and the measure of angle π΄πΈπΆ as π¦ degrees. In the second part of the
question, we identified another pair of angle measures that added to 180
degrees. And since π₯ degrees plus π¦
degrees equals 180 degrees, then we can say that the measure of angle πΆπΈπ΅ is
also π₯ degrees.
So, we can say that the
statement that the measure of angle π΄πΈπ· equals the measure of angle πΆπΈπ΅ 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 π΄πΈπΆ equals the measure
of angle π·πΈπ΅. These vertically opposite
angles will both be π¦ 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. Weβll 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. Letβs 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, letβs say that the two
parallel lines are the lines π΄π΅ and πΆπ· and the line which is perpendicular
to line π΄π΅ is the line πΈπΉ. We can also label the point
where line π΄π΅ and πΈπΉ intersect as π and where line πΆπ· and πΈπΉ intersect
as point π. 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 weβve
called line πΆπ·, also perpendicular to line πΈπΉ? Since we know that we have this
relationship where the two lines are perpendicular, we can write that the
measure of angle πΈππ΅ is 90 degrees. Then, we can use the properties
of parallel lines to help us with another fact. Angles πΈππ΅ and πΈππ· 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 πΈπΉ is also
perpendicular to line πΆπ·. 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. Letβs 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, weβll 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 π΅πΆπ·.
So, letβs 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 π΄π΅ and π΄π· are
congruent, and sides πΆπ΅ and πΆπ· are congruent. We also have the angle measure
of angle π΄πΆπ· given as 29 degrees. Since we are told to use the
properties of congruent triangles, letβs identify two triangles we may be able
to use.
We have triangle π΄π΅πΆ and
triangle π΄π·πΆ. These two triangles share a
common side of π΄πΆ. 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
π΄π΅πΆ is congruent to triangle π΄π·πΆ 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 π΄πΆπ΅ and
π΄πΆπ· are corresponding, these will both be 29 degrees. But we are asked to find the
measure of angle π΅πΆπ·. Since angle π΅πΆπ· is comprised
of angles π΄πΆπ΅ and π΄πΆπ·, 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 π΄πΆπ΅ and π΄πΆπ·, we could just have easily
had a general angle of π₯ 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 π₯
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.