Question

2-5 study guide and intervention
postulates and paragraph proofs
points, lines, and planes in geometry, a postulate is a statement that is accepted as true. postulates describe fundamental relationships in geometry.
postulate: through any two points, there is exactly one line.
postulate: through any three points not on the same line, there is exactly one plane.
postulate: a line contains at least two points.
postulate: a plane contains at least three points not on the same line.
postulate: if two points lie in a plane, then the line containing those points lies in the plane.
postulate: if two lines intersect, then their intersection is exactly one point.
postulate: if two planes intersect, then their intersection is a line.

exercises
use postulates to determine whether each statement is always, sometimes, or never true.

1. a line contains exactly one point.
2. noncollinear points \\(r\\), \\(s\\), and \\(t\\) are contained in exactly one plane.
3. any two lines \\(\ell\\) and \\(m\\) intersect.
4. if points \\(g\\) and \\(h\\) are contained in plane \\(\mathcal{m}\\), then \\(\overline{gh}\\) is perpendicular to plane \\(\mathcal{m}\\).
5. planes \\(\mathcal{r}\\) and \\(\mathcal{s}\\) intersect in point \\(t\\).
6. if points \\(a\\), \\(b\\), and \\(c\\) are noncollinear, then segments \\(\overline{ab}\\), \\(\overline{bc}\\), and \\(\overline{ca}\\) are contained in exactly one plane.

in the figure, \\(\overleftrightarrow{ac}\\) and \\(\overleftrightarrow{de}\\) are in plane \\(\mathcal{q}\\) and \\(\overleftrightarrow{ac} \parallel \overleftrightarrow{de}\\).
state the postulate that can be used to show each statement is true.

1. exactly one plane contains points \\(f\\), \\(b\\), and \\(e\\).
2. \\(\overleftrightarrow{be}\\) lies in plane \\(\mathcal{q}\\).

Explanation:

Analyze Exercises 1 to 6

We evaluate each statement using the provided geometric postulates:

• Exercise 1: "A line contains exactly one point."
• Postulate: "A line contains at least two points."
• Since a line must have at least two points, it can never contain exactly one point.
• Result: Never

• Exercise 2: "Noncollinear points \(R\), \(S\), and \(T\) are contained in exactly one plane."
• Postulate: "Through any three points not on the same line, there is exactly one plane."
• Noncollinear means not on the same line.
• Result: Always

• Exercise 3: "Any two lines \(\ell\) and \(m\) intersect."
• In geometry, two lines can be parallel or skew (not intersecting).
• Result: Sometimes

• Exercise 4: "If points \(G\) and \(H\) are contained in plane \(\mathcal{M}\), then \(\overline{GH}\) is perpendicular to plane \(\mathcal{M}\)."
• Postulate: "If two points lie in a plane, then the line containing those points lies in the plane."
• If the line lies entirely within the plane, it cannot be perpendicular to it (unless it is a single point, which a line is not).
• Result: Never

• Exercise 5: "Planes \(\mathcal{R}\) and \(\mathcal{S}\) intersect in point \(T\)."
• Postulate: "If two planes intersect, then their intersection is a line."
• Two planes cannot intersect in only a single point.
• Result: Never

• Exercise 6: "If points \(A\), \(B\), and \(C\) are noncollinear, then segments \(\overline{AB}\), \(\overline{BC}\), and \(\overline{CA}\) are contained in exactly one plane."
• Since \(A\), \(B\), and \(C\) are noncollinear, they define exactly one plane.
• Any line (or segment) connecting these points must lie entirely within that same plane.
• Result: Always

Analyze Exercise 7

• Statement: "Exactly one plane contains points \(F\), \(B\), and \(E\)."
• Analysis: Looking at the figure, points \(F\), \(B\), and \(E\) are three noncollinear points.
• Postulate: "Through any three points not on the same line, there is exactly one plane." (or "Through any three noncollinear points, there is exactly one plane.")

Analyze Exercise 8

• Statement: "\(\overleftrightarrow{BE}\) lies in plane \(\mathcal{Q}\)."
• Analysis: Points \(B\) and \(E\) both lie in plane \(\mathcal{Q}\).
• Postulate: "If two points lie in a plane, then the line containing those points lies in the plane."

Answer:

Question 1

Never

Question 2

Always

Question 3

Sometimes

Question 4

Never

Question 5

Never

Question 6

Always

Question 7

Through any three points not on the same line, there is exactly one plane.

Question 8

If two points lie in a plane, then the line containing those points lies in the plane.