Introduction to Geometry (Level 3)

In this video we are going to practice denoting and naming angles by using various

points located on a geometric figure. Let’s go ahead and jump straight into the first

example. Using the following figure find the following.

Name angle OPR in all other possible ways. The following figure is a type of quadrilateral

since it is formed by 4 sides as oppose to a triangle which is formed by 3 sides. We

will study quadrilaterals in a much later video for now the problem is asking us to

name angle OPR in all other possible ways. Lets first figure out where angle OPR is located

on the figure, so we first locate point O and move towards point P and then we move

towards point R, from the way the angle is denoted we know that the vertex of the angle

is located at point P and is formed by rays PO and PR, the angle in question would be the following the highlighted angle. Now you might be tempted to use vertex P and name this angle as, angle

P. Unfortunately, this would be incorrect since point P is also the vertex of another

angle specifically this one, so we can’t name angle OPR as angle P since this point

is also the vertex of another angle, this would be confusing to someone who is trying to locate

angle OPR, in an effort to avoid any confusion we need to name this angle in other ways.

One alternative way we can name this angle is by reversing the order of the points and

name it as angle RPO making sure we denote the vertex P in the middle. Yet another way

to name this angle is by using point S since it is located in same line containing points

P and O, so we can name this angle as angle SPR or angle RPS making sure we denote the vertex

P in the middle. Notice that the figure does not use numbers to label the angles as a result

we are essentially done coverinig all the possible ways to name this particular angle.

Alright let’s try the next example. What is the vertex of angle TOS.

Remember the vertex of an angle when denoted in this particular way is represented by the

middle letter; in this case the vertex of this angle is point O. In the figure to the

left this is the location where rays OT and OS intersect you can also think about a vertex

as the intersection of these two rays. Using set notation we can denote them as Ray OT

intersected with Ray OS. We will cover problems involving set notation

in a later video. Now let’s move along to the next example.

How many angles have as its vertex point R? First let’s locate point R, point R is located

on the bottom right corner of the figure. Notice that this point is the vertex of three

distinct angles the first angle can be named as: angle PRO or angle ORP or angle PRT or

angle TRP the second angle can be named as angle ORS or angle SRO or angle SRT or angle

TRS the third and final angle can be named as: angle PRS or angle SRP so in total there

are 3 angles that have point R as its vertex. Alright let’s try the next example.

Name angle TSP in all other possible ways. Ok let’s first identify what angle the problem

is referring to, so let’s start at point T and move towards point S and then move towards

point P. So this is the angle referenced by the problem. One way we can name this angle

is by switching the direction of the points and name it as: angle PST, in addition we can also

use point O to name this angle since its located on the same line that contains point

P and S, using this point we name this angle as: angle OST or angle TSO, once again since this figure does not use numbers to label its angles these are essentially all the possible ways to name

angle TSP. Alright Let’s try the final example. How many triangles are there in the figure?

At first glance it seems that there are 4 triangles in the figure, they include the

triangle on the left which can be named as triangle TOP, the triangle on the right which

can be named as triangle SOR, the triangle on the top which can be named as triangle

TOS and the triangle on the bottom which can be named as triangle POR. It turns out that

there are actually other triangles that are formed by these 4 smaller triangles, the first

one can be named as triangle PTS the next one can be named as triangle TRS, another

one can be named as triangle SRP and the last one can be named as triangle RPT. So there

a total of 8 triangles in this figure. Alright in our next video we will continue

denoting and naming more challenging angles by using the points located on a geometric

figure.