Often, we want to ‘just capture’ an object inside an angle, relative to some external ‘viewing point’.
The word ‘subtend’ is used in this context, as the examples below illustrate:
the angle subtended by a tree at a point 30 feet from the base of the tree 
the angle subtended by the moon at a point on Earth (not to scale) 
the angle ($\,\theta\,$) subtended by an arc (at the center of the circle) 
the angle subtended by an arc at a noncenter point 
The use of the word ‘subtend’ is a bit murky in the literature.
Rarely do people seem to define it; when an attempt is made, it is usually for a specific occurrence of the idea.
My genius husband (Ray) and I had fun defining the concept for a
very general situation. Enjoy!
First, a motivational discussion from a discipline that uses the word ‘subtend’ a lot! Note: When a degree is divided into $\,60\,$ equal parts, each part is called a ‘minute’. The symbol $'$ is used for minutes: $\,1' = (\frac{1}{60})^\circ\,$

an eye chart 
Some preliminary definitions:




Take a look at the interval $\,[3,4)\,$ below: notice that the right endpoint isn't included.
Let $\,S\,$ be a nonempty set of real numbers.

some upper bounds for the set $\,[3,4)\,$ 


Here's the rough idea:
Humans are usually good at detecting the ‘edge’ or ‘boundary’ of geometric objects,
but it's a bit tricky to define mathematically.
(Topology can do it quite easily, but that's beyond the scope of this course.)
The problem is that a boundary point of an object might
be part of the object, or not.
(For example, in the interval $\,[3,4)\,,$ the number $\,4\,$ is a boundary point,
but isn't in the interval.)
Pay attention to how the least upper bound is used to resolve this issue in the definition below.
Now, let's make this idea mathematically precise:

In all these examples:
a point $\alpha = 0^\circ$, $\beta = 0^\circ$ $\theta_{\mathcal G, P} = 0^\circ$ 
a line segment $\alpha = 0^\circ$, $\beta = 0^\circ$ $\theta_{\mathcal G, P} = 0^\circ$ 
a line segment $\alpha = 52^\circ$, $\beta = 0^\circ$ $\theta_{\mathcal G, P} = 52^\circ$ When sweeping clockwise, $\,\theta_{\text{sweep}}\,$ takes on values in the interval $\,[0^\circ,52^\circ)\,$; the supremum is $\,52^\circ\,$ 
a line segment $\alpha = 0^\circ$, $\beta = 52^\circ$ $\theta_{\mathcal G, P} = 52^\circ$ When sweeping counterclockwise, $\,\theta_{\text{sweep}}\,$ takes on values in the interval $\,[0^\circ,52^\circ]\,$; the supremum is $\,52^\circ\,$ 
a line segment $\alpha = 26^\circ$, $\beta = 26^\circ$ $\theta_{\mathcal G, P} = 52^\circ$ 
a circle $\theta_{\mathcal G, P} = 360^\circ$ 
a punctured circle $\alpha = 90^\circ$, $\beta = 270^\circ$ $\theta_{\mathcal G, P} = 360^\circ$ 
a circle $\alpha = 33^\circ$, $\beta = 0^\circ$ $\theta_{\mathcal G, P} = 33^\circ$ 
a filled polygon $\alpha = 0^\circ$, $\beta = 103^\circ$ $\theta_{\mathcal G, P} = 103^\circ$ 
a filled polygon $\alpha = 57^\circ$, $\beta = 49^\circ$ $\theta_{\mathcal G, P} = 106^\circ$ 

Using the original definition Using the equivalent definition 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
