Question

post test: foundations of geometry
diagrams with labels c, d, t, m; symbols: $overrightarrow{cd}$, $overline{cd}$, $angle ctm$ with boxes to match (diagrams include segments, rays, and angle diagrams)

Answer:

To solve this problem, we analyze each geometric notation and match it to the corresponding diagram:

1. For \(\overrightarrow{CD}\) (Ray \(CD\)):

A ray has a starting point and extends infinitely in one direction. \(\overrightarrow{CD}\) starts at \(C\) and goes through \(D\) (extending beyond \(D\)). Looking at the diagrams, the top - left and middle - right diagrams have a ray starting at \(C\) and passing through \(D\) (with an arrow at \(D\)’s end to show infinite extension). But the middle - right diagram (the one with \(C\) and \(D\) on a line, \(C\) on the left, \(D\) on the right, and the line from \(C\) through \(D\) with an arrow at \(D\)) is a better match for a ray. Wait, actually, the top - left: let's re - examine. The first row, first diagram: \(C\) and \(D\) with a ray? No, the middle - right diagram (the one with \(C\) and \(D\) where the line is from \(C\) to \(D\) with an arrow at \(D\)) is a ray. Wait, also, the first row, third diagram? No, let's think again. The notation \(\overrightarrow{CD}\) means the ray starts at \(C\) and goes towards \(D\) (and beyond). So the diagram where there is a ray starting at \(C\), passing through \(D\), and extending past \(D\) is the one with \(C\) and \(D\) on a horizontal line, \(C\) left, \(D\) right, and the arrow at \(D\)’s end. Among the given diagrams, the middle - right diagram (the blue - white one with \(C\) and \(D\) and the arrow at \(D\)) or the top - left? Wait, the top - left has \(C\) and \(D\) with a ray? Wait, the problem's diagrams: let's list the diagrams:

• Top row:
• First: \(C\) and \(D\) on a line, blue below, arrow at \(D\)’s end (ray \(CD\))?
• Second: \(T\), \(M\) with two rays from \(T\) to \(M\) and another?
• Third: \(C\) and \(D\) with blue above, arrow at \(D\)’s end?

• Middle row:
• First: \(C\), \(T\), \(M\) with two rays from \(C\) to \(T\) and \(C\) to \(M\) (angle diagram)
• Second: \(C\) and \(D\) with a line segment (since no arrow beyond \(D\))? Wait, no, the middle - right diagram: \(C\) and \(D\) with a line from \(C\) to \(D\) and no arrow beyond? Wait, maybe I mis - see. Let's correct:

\(\overrightarrow{CD}\) is a ray. So it should have a starting point \(C\) and an arrow indicating it goes through \(D\) to infinity. So the diagrams with a ray starting at \(C\), passing through \(D\), and with an arrow at \(D\)’s end. So the top - left and top - third? Wait, no, the middle - right diagram: if it's a line segment, then no. Wait, the first row, first diagram: \(C\) and \(D\) with a ray (arrow at \(D\)). The middle - right diagram: \(C\) and \(D\) with a line (no arrow beyond \(D\))? Wait, maybe the middle - right is a line segment. Wait, the notation \(\overline{CD}\) is a line segment. So:

2. For \(\overline{CD}\) (Line Segment \(CD\)):

A line segment has two endpoints \(C\) and \(D\) and no infinite extension. So the diagram where \(C\) and \(D\) are connected by a line with no arrows beyond (just between \(C\) and \(D\)). The middle - right diagram (the blue - white one with \(C\) and \(D\) and a line between them, no arrows beyond) is a line segment.

3. For \(\angle CTM\) (Angle at \(T\) with sides \(TC\) and \(TM\)):

An angle is formed by two rays with a common vertex. Here, the vertex is \(T\), and the sides are \(TC\) and \(TM\). So the diagram with \(T\) as the vertex, and two rays from \(T\) (or from \(C\) to \(T\) and \(M\) to \(T\)?) Wait, \(\angle CTM\) has vertex \(T\), so the sides are \(TC\) and \(TM\). So the diagram where \(T\) is the vertex, and there are two rays: one from \(T\) to \(C\) and one from \(T\) to \(M\) (or from \(C\) to \(T\) and \(M\) to \(T\)). Looking at the diagrams, the top - second diagram (with \(T\) and \(M\), and a ray from \(T\) to \(M\) and another from \(T\) to \(C\)?) or the middle - first diagram (with \(C\), \(T\), \(M\), and two rays from \(C\) to \(T\) and \(C\) to \(M\))? Wait, no: \(\angle CTM\) has vertex \(T\), so the angle is at \(T\). So the top - second diagram: \(T\) is the vertex, with a ray to \(M\) and a ray to \(C\), forming an angle at \(T\).

Matching:

• \(\overrightarrow{CD}\) (Ray \(CD\)): The diagram with \(C\) and \(D\) where there is a ray starting at \(C\), passing through \(D\), and with an arrow at \(D\)’s end (e.g., top - left or top - third? Wait, the first row, first diagram: \(C\) and \(D\) with a ray (arrow at \(D\)). Let's assume the first row, first diagram is for \(\overrightarrow{CD}\).
• \(\overline{CD}\) (Line Segment \(CD\)): The middle - right diagram (with \(C\) and \(D\) connected by a line, no arrows beyond).
• \(\angle CTM\) (Angle at \(T\)): The top - second diagram (with \(T\) as the vertex, and two rays forming the angle).

Since the problem is about matching the geometric notations to the diagrams, and if we have to box the answers (but the original problem is a matching), but if we assume the diagrams are labeled as follows (from top - left, top - middle, top - right, middle - left, middle - right):

1. Top - left: Ray \(CD\) (matches \(\overrightarrow{CD}\))
2. Top - middle: \(\angle CTM\) (matches \(\angle CTM\))
3. Middle - right: Line Segment \(CD\) (matches \(\overline{CD}\))

So the matching is:

• \(\overrightarrow{CD}\) → Top - left (or the diagram with ray \(CD\))
• \(\overline{CD}\) → Middle - right (or the diagram with line segment \(CD\))
• \(\angle CTM\) → Top - middle (or the diagram with angle at \(T\))

If we have to present the answer in a boxed form (assuming the diagrams are numbered 1 - 5 from top - left to middle - right):

• \(\overrightarrow{CD}\): Let's say diagram 1 (top - left)
• \(\overline{CD}\): Diagram 5 (middle - right)
• \(\angle CTM\): Diagram 2 (top - middle)

But since the original problem is a matching, the final answer (matching) would be:

\(\overrightarrow{CD}\) matches the diagram with ray \(CD\) (e.g., top - left), \(\overline{CD}\) matches the diagram with line segment \(CD\) (middle - right), and \(\angle CTM\) matches the diagram with angle at \(T\) (top - middle).

If we have to box each match:

For \(\overrightarrow{CD}\): \(\boxed{\text{Top - left diagram (ray } CD\text{)}}\)
For \(\overline{CD}\): \(\boxed{\text{Middle - right diagram (line segment } CD\text{)}}\)
For \(\angle CTM\): \(\boxed{\text{Top - middle diagram (angle } CTM\text{)}}\)