Draw Rough Diagrams of Two Angles with Common Points

To demonstrate how two angles can share common points, let’s break it down according to the number of common points.

a. One Point Common

Here, we have two angles that meet at a single point. For instance, angle AOB and angle COD both share point O. The diagrams would look something like this:

     A
     |\
     | \
     |  \    C
     |   \   /|
     |    \ / |
     O--------D
      \
       \
        B

b. Two Points in Common

In this scenario, the two angles share two points. For instance, angle AOB and angle COD could meet at points O and P. The rough diagram may appear like this:

     A
     |\
     | \ 
     |  \ P
     O----K----D
     |  /       |
     | /        |
     B          C

c. Three Points in Common

Two angles sharing three points could look like this. For example, angles AOB and COD intersect at points O, P, and Q:

     A
     |\   Q
     | \  / 
     |  \/   C
     O--------D
         |  / 
         | / 
         P
         | 
         B

d. Four Points in Common

When two angles share four points, it may look something like the following. Assuming points O, P, Q, and R are common:

       A
       |\
       | \ P
       |  \  |
       O----D----Q
        |  /|   |
        | / | R 
        B   | 

e. One Ray in Common

For angles sharing just one ray, consider angles AOB and AOC. They share the ray OA while diverging towards B and C:

       A
       |\
       | \
       |  \
       O---B
       |  
       |    C

Each of these diagrams represents how two angles can share common points or portions, showcasing the varied relationships between angles in geometry.