1. **Problem 1: Prove every segment has a unique midpoint.** A midpoint $M$ of segment $AB$ is a point on $AB$ such that $AM = MB$. Since a segment is part of a line, and a line extends infinitely in both directions, there must be exactly one point $M$ on $AB$ that divides it into two equal parts. Uniqueness follows because if there were two distinct midpoints, say $M$ and $N$, then $AM = MB$ and $AN = NB$, but $M \neq N$ contradicts the segment length equality. 2. **Problem 2: Show that if $A \cdot B \cdot C$ and $B \cdot C \cdot D$, then $A \cdot B \cdot D$ and $A \cdot C \cdot D$.** Given points $A, B, C, D$ on a line with $B$ between $A$ and $C$, and $C$ between $B$ and $D$. From $A \cdot B \cdot C$, $B$ lies between $A$ and $C$. From $B \cdot C \cdot D$, $C$ lies between $B$ and $D$. Therefore, the order on the line is $A - B - C - D$. Hence, $A \cdot B \cdot D$ (since $B$ is between $A$ and $D$) and $A \cdot C \cdot D$ (since $C$ is between $A$ and $D$). 3. **Problem 3: Define the term closed half plane $CHP(l,P)$.** A closed half plane $CHP(l,P)$ determined by a line $l$ and a point $P$ not on $l$ is the set of all points on $l$ and on the same side of $l$ as $P$. It includes the line $l$ itself and all points on one side of $l$. 4. **Problem 4: Let rays $\vec{a}$ and $\vec{b}$ be given. Prove that the angle $\angle ab = 0$ if and only if the angle $\angle ab$ is the zero angle.** By definition, the angle between two rays is zero if and only if the rays coincide or lie on the same line with the same direction. If $\angle ab = 0$, then rays $\vec{a}$ and $\vec{b}$ overlap exactly, so the angle is the zero angle. Conversely, if the angle is the zero angle, then the rays coincide, so $\angle ab = 0$. 5. **Problem 5: State and prove the Angle Construction Lemma.** **Statement:** Given a ray $\vec{a}$ and an angle measure $\theta$, there exists a unique ray $\vec{b}$ such that $\angle ab = \theta$. **Proof:** 1. Start with ray $\vec{a}$ and point $O$ as the vertex. 2. Using a protractor or angle measure, mark point $P$ such that $\angle aOP = \theta$. 3. Draw ray $\vec{b}$ from $O$ through $P$. 4. Uniqueness follows because any other ray forming the same angle $\theta$ with $\vec{a}$ must coincide with $\vec{b}$. Thus, the angle construction lemma is proved. **Final answers:** - Every segment has a unique midpoint. - If $A \cdot B \cdot C$ and $B \cdot C \cdot D$, then $A \cdot B \cdot D$ and $A \cdot C \cdot D$. - Closed half plane $CHP(l,P)$ is the set of points on line $l$ and on the same side as $P$. - $\angle ab = 0$ if and only if $\angle ab$ is the zero angle. - Angle construction lemma states existence and uniqueness of a ray forming a given angle with another ray.