Question

for use with the lesson \identify points, lines, and planes\use the diagram to decide whether the given statement is true or false.1. points h, i, and g are collinear.2. points h, i, and j are coplanar.3. $overrightarrow{eg}$ and $overrightarrow{fg}$ are opposite rays.4. all points on $overrightarrow{gi}$ and $overrightarrow{gf}$ are coplanar.5. the intersection of $overrightarrow{ef}$ and plane jkh is $overrightarrow{hi}$.6. the intersection of $overrightarrow{ef}$, $overrightarrow{hi}$, and $overrightarrow{jg}$ is point g.7. the intersection of plane egh and plane jgi is point g.8. the intersection of plane efi and plane jkg is $overrightarrow{hg}$.

Explanation:

Step1: Evaluate collinearity (1)

Collinear points lie on one line. Points $H, I, G$ lie on a single line.

Step2: Evaluate coplanarity (2)

Any 3 points are coplanar. $H, I, J$ form a plane.

Step3: Evaluate opposite rays (3)

Opposite rays share endpoint, go opposite. $\overrightarrow{EG}$ and $\overrightarrow{FG}$ share $G$, extend opposite.

Step4: Evaluate ray coplanarity (4)

$\overrightarrow{GI}$ and $\overrightarrow{GF}$ share $G$, lie on one plane.

Step5: Evaluate line-plane intersection (5)

$\overleftrightarrow{EF}$ is parallel to plane $JKH$, no intersection with $\overleftrightarrow{HI}$.

Step6: Evaluate 3-line intersection (6)

$\overleftrightarrow{EF}$, $\overleftrightarrow{HI}$, $\overleftrightarrow{JG}$ all meet only at $G$.

Step7: Evaluate plane-plane intersection (7)

Two planes intersect in a line, not a single point.

Step8: Evaluate plane-plane intersection (8)

Plane $EFI$ and plane $JKG$ intersect at $\overleftrightarrow{HG}$.

Answer:

1. True
2. True
3. True
4. True
5. False
6. True
7. False
8. True