Question

lesson 1.1 practice a for use with the lesson \identify points, lines, and planes\
in exercises 1 - 8, use the diagram.

1. give two other names for $overleftrightarrow{ab}$.
2. name three points that are collinear.
3. give another name for plane $f$.
4. name a point that is not coplanar with $a$, $b$, and $c$.
5. give another name for $overline{cd}$.
6. name three rays with endpoint $b$.
7. name a pair of opposite rays.
8. give another name for $overrightarrow{cd}$.

sketch the figure described.

1. three points that are collinear
2. four points that are coplanar
3. three lines that intersect at one point
4. a line and a plane that intersect at one point

in exercises 13 - 20, use the diagram.

1. are points $j$, $k$, and $l$ collinear?
2. are points $j$, $k$, and $l$ coplanar?
3. are points $j$, $k$, and $m$ collinear?
4. are points $j$, $k$, and $m$ coplanar?
5. name the intersection of $overleftrightarrow{kl}$ and $overleftrightarrow{pq}$.
6. name the intersection of $overleftrightarrow{pq}$ and plane $kmn$.
7. name the intersection of plane $r$ and plane $s$.
8. name three pairs of opposite rays.

Explanation:

Step1: Recall line - naming rules

A line can be named by any two points on it. For $\overleftrightarrow{AB}$, two other names could be $\overleftrightarrow{BA}$ and any other two - point combination on the same line (assuming there are more points). Let's say if there is a point $X$ on the line, it could also be named $\overleftrightarrow{AX}$ or $\overleftrightarrow{BX}$. For simplicity, $\overleftrightarrow{BA}$ and assume another point $E$ on the line, $\overleftrightarrow{AE}$.

Step2: Recall collinear - points concept

Collinear points lie on the same line. Looking at the first diagram, points $A$, $B$, and $H$ are on the same line.

Step3: Recall plane - naming rules

A plane can be named by three non - collinear points on it. For plane $F$, another name could be plane $AEB$ (assuming $E$ is a non - collinear point with $A$ and $B$ on the plane).

Step4: Recall coplanar - points concept

Coplanar points lie on the same plane. Point $D$ is not on the same plane as $A$, $B$, and $C$.

Step5: Recall line - segment naming rules

A line segment $\overline{CD}$ can also be named $\overline{DC}$.

Step6: Recall ray - naming rules

Rays with endpoint $B$ are $\overrightarrow{BA}$, $\overrightarrow{BH}$, $\overrightarrow{BC}$.

Step7: Recall opposite - rays concept

Opposite rays are two rays that have the same endpoint and form a straight line. $\overrightarrow{BC}$ and $\overrightarrow{BD}$ are opposite rays.

Step8: Similar to step5

Another name for $\overrightarrow{CD}$ is $\overrightarrow{DC}$.

Step9: Sketching collinear points

Draw three points on a straight line.

Step10: Sketching coplanar points

Draw four points on a flat surface (like a piece of paper representing a plane).

Step11: Sketching intersecting lines

Draw three lines that meet at a single point.

Step12: Sketching line - plane intersection

Draw a line that pierces a plane at one point.

Step13: Check collinearity

By looking at the second diagram, points $J$, $K$, and $L$ are not collinear as they do not lie on the same straight line.

Step14: Check coplanarity

Points $J$, $K$, and $L$ are coplanar as they can be considered to lie on the same plane.

Step15: Check collinearity

Points $J$, $K$, and $M$ are not collinear.

Step16: Check coplanarity

Points $J$, $K$, and $M$ are coplanar.

Step17: Find line - line intersection

The intersection of $\overleftrightarrow{KL}$ and $\overleftrightarrow{PQ}$ is the point where they cross. Let's assume it is point $N$ (from the diagram).

Step18: Find line - plane intersection

The intersection of $\overleftrightarrow{PQ}$ and plane $KMN$ is the point where the line enters the plane. Let's assume it is point $N$.

Step19: Find plane - plane intersection

The intersection of plane $R$ and plane $S$ is a line. Let's call it line $\overleftrightarrow{MN}$ (assuming $M$ and $N$ are two points on the intersection line).

Step20: Find opposite rays

Three pairs of opposite rays are: $\overrightarrow{KM}$ and $\overrightarrow{KN}$, $\overrightarrow{MN}$ and $\overrightarrow{MK}$, $\overrightarrow{NK}$ and $\overrightarrow{NM}$.

Answer:

1. $\overleftrightarrow{BA}$, $\overleftrightarrow{AE}$
2. $A$, $B$, $H$
3. plane $AEB$
4. $D$
5. $\overline{DC}$
6. $\overrightarrow{BA}$, $\overrightarrow{BH}$, $\overrightarrow{BC}$
7. $\overrightarrow{BC}$ and $\overrightarrow{BD}$
8. $\overrightarrow{DC}$
9. (Sketch of three points on a line)
10. (Sketch of four points on a plane)
11. (Sketch of three lines intersecting at a point)
12. (Sketch of a line intersecting a plane at a point)
13. No
14. Yes
15. No
16. Yes
17. Point $N$
18. Point $N$
19. Line $\overleftrightarrow{MN}$
20. $\overrightarrow{KM}$ and $\overrightarrow{KN}$, $\overrightarrow{MN}$ and $\overrightarrow{MK}$, $\overrightarrow{NK}$ and $\overrightarrow{NM}$