Question

in exercises 13–15, use the diagram.

1. name 12 different rays.
2. name a pair of opposite rays.
3. name 3 lines that intersect at point c.

Explanation:

Question 13: Name 12 different rays.

Step 1: Recall the definition of a ray

A ray is a part of a line that starts at a point (endpoint) and extends infinitely in one direction. So, for each line and each point on the line, we can define rays starting at that point and going along the line.

Step 2: Identify points and lines in the diagram

The diagram has points A, B, C, D, E with lines: one horizontal line through A and B (let's call it line AB), one horizontal line through D and E (line DE), and two lines intersecting at C (let's say line CD and line CE, and line AB also passes through C).

Step 3: List rays from each point

• From point C:
• Along line AB: \( \overrightarrow{CA} \) (starts at C, goes through A) and \( \overrightarrow{CB} \) (starts at C, goes through B).
• Along line CD: \( \overrightarrow{CD} \) (starts at C, goes through D) and \( \overrightarrow{DC} \) (starts at D, goes through C? Wait, no: ray starting at D, going through C is \( \overrightarrow{DC} \); ray starting at C, going through D is \( \overrightarrow{CD} \).
• Along line CE: \( \overrightarrow{CE} \) (starts at C, goes through E) and \( \overrightarrow{EC} \) (starts at E, goes through C).
• From point D:
• Along line DE: \( \overrightarrow{DE} \) (starts at D, goes through E) and \( \overrightarrow{ED} \) (starts at E, goes through D).
• Along line CD: \( \overrightarrow{DC} \) (starts at D, goes through C) and \( \overrightarrow{CD} \) (starts at C, goes through D) – but we already listed \( \overrightarrow{CD} \) from C.
• From point E:
• Along line DE: \( \overrightarrow{ED} \) (starts at E, goes through D) and \( \overrightarrow{DE} \) (starts at D, goes through E) – already listed.
• Along line CE: \( \overrightarrow{EC} \) (starts at E, goes through C) and \( \overrightarrow{CE} \) (starts at C, goes through E) – already listed.
• From point A:
• Along line AB: \( \overrightarrow{AC} \) (starts at A, goes through C) and \( \overrightarrow{AB} \) (but B is on the line, so \( \overrightarrow{AB} \) is same as \( \overrightarrow{ACB} \), but we can use \( \overrightarrow{AC} \) (starts at A, through C) and \( \overrightarrow{AB} \) (starts at A, through B) – but \( \overrightarrow{AB} \) is same as \( \overrightarrow{ACB} \), so better to use \( \overrightarrow{AC} \) (starts at A, through C) and \( \overrightarrow{AB} \) (starts at A, through B). Wait, but \( \overrightarrow{CA} \) is from C to A, \( \overrightarrow{AC} \) is from A to C.
• From point B:
• Along line AB: \( \overrightarrow{BC} \) (starts at B, through C) and \( \overrightarrow{BA} \) (starts at B, through A) – but \( \overrightarrow{BA} \) is same as \( \overrightarrow{BCA} \), so \( \overrightarrow{BC} \) (starts at B, through C) and \( \overrightarrow{BA} \) (starts at B, through A).

Combining these, a set of 12 rays could be: \( \overrightarrow{CA} \), \( \overrightarrow{CB} \), \( \overrightarrow{CD} \), \( \overrightarrow{CE} \), \( \overrightarrow{AC} \), \( \overrightarrow{BC} \), \( \overrightarrow{DC} \), \( \overrightarrow{EC} \), \( \overrightarrow{DE} \), \( \overrightarrow{ED} \), \( \overrightarrow{AD} \) (assuming line AD exists), \( \overrightarrow{DA} \) (assuming line DA exists). The key is to ensure each ray has a unique endpoint and direction.

Question 14: Name a pair of opposite rays.

Step 1: Recall the definition of opposite rays

Opposite rays are two rays that share the same endpoint and form a straight line (i.e., they are collinear and point in opposite directions).

Step 2: Identify collinear rays with the same endpoint

In the diagram, points A, C, B are collinear (on a straight horizontal line). So, the ray starting at C and going through A (\( \overrightarrow{CA} \)) and the ray starting at C and going through B (\( \overrightarrow{CB} \)) share the same endpoint C and form a straight line (line AB), so they are opposite rays.

Question 15: Name 3 lines that intersect at point C.

Step 1: Recall the definition of a line

A line is a straight, infinite path with no endpoints.

Step 2: Identify lines passing through point C

From the diagram, point C is the intersection of:

• The horizontal line passing through A and B (line \( AB \)).
• A line passing through C and D (line \( CD \)).
• A line passing through C and E (line \( CE \)).

These three lines all intersect at point C.

Answer:

Ray \( \overrightarrow{CA} \), Ray \( \overrightarrow{CB} \), Ray \( \overrightarrow{CD} \), Ray \( \overrightarrow{CE} \), Ray \( \overrightarrow{DC} \), Ray \( \overrightarrow{ED} \), Ray \( \overrightarrow{DE} \), Ray \( \overrightarrow{AC} \), Ray \( \overrightarrow{BC} \), Ray \( \overrightarrow{EC} \), Ray \( \overrightarrow{AD} \) (assuming a line through A and D), Ray \( \overrightarrow{DA} \) (assuming a line through D and A) (Note: The exact 12 rays depend on the diagram's lines, but a typical set from the given points A, B, C, D, E with lines through them would include rays from each intersection point along the lines. For example, from point C: \( \overrightarrow{CA} \), \( \overrightarrow{CB} \), \( \overrightarrow{CD} \), \( \overrightarrow{CE} \); from point D: \( \overrightarrow{DE} \), \( \overrightarrow{DC} \), \( \overrightarrow{DA} \) (if line AD exists), \( \overrightarrow{DB} \) (if line DB exists); from point E: \( \overrightarrow{ED} \), \( \overrightarrow{EC} \), \( \overrightarrow{EA} \) (if line EA exists), \( \overrightarrow{EB} \) (if line EB exists). A common valid set could be \( \overrightarrow{CA} \), \( \overrightarrow{CB} \), \( \overrightarrow{CD} \), \( \overrightarrow{CE} \), \( \overrightarrow{AC} \), \( \overrightarrow{BC} \), \( \overrightarrow{DC} \), \( \overrightarrow{EC} \), \( \overrightarrow{DE} \), \( \overrightarrow{ED} \), \( \overrightarrow{AD} \), \( \overrightarrow{DA} \) (adjusted based on the diagram's lines).)