Introduction to Geometry (Level 3) | Naming Angles I

Introduction to Geometry (Level 3) | Naming Angles I

Alright in this post 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.

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 post 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 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 some 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, 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.

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 angel 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:

We will cover problems involving set notation in a later post. Now let’s move along to the next example.

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.

Let’s move along to the next example.

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 it 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.

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 post we will continue denoting and naming more challenging angles by using the points located on a geometric figure.